1 / 28

cliqueTrees

In taking the inner product of 32 bitwidth Scalar pTreeSets (e.g., for Oblique or Hull Classification) we want line segments to be tight against the Training Class, but not too tight (because Training Classes are almost always only estimates of the actual classes). 

efugate
Download Presentation

cliqueTrees

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. In taking the inner product of 32 bitwidth Scalar pTreeSets (e.g., for Oblique or Hull Classification) we want line segments to be tight against the Training Class, but not too tight (because Training Classes are almost always only estimates of the actual classes).  I.e., we may want to leave room between the Training Class and the bordering line segment, because of the approximate-ness of the Training Classes We can do that as follows: For the segment on the Minimum side (the segment perpendicular to the unit vector, d, through minimum{d dot x | xTraining Class}, set the 24 LoBits to 0 (only the 8 HiBits are then used in inner product). This moves the bordering line segment away from that Training Class on that side. For the segment on the Maximum side set the 24 LoBits to 1 (Better yet, add 1 to Hi 8th bits, i.e., set the 24 LoBits=0, add 1 to the 8th HiBit (which is almost the same as setting the 24 LoBits to all 1s but gives a much faster inner product calculation). This moves the bordering line segment away from that Training Class on the other side. This approach is is a win-win: it places the line segments better for Classification, and it lowers inner product costs  (to 8 bit-width costs, instead of 32 bit-width costs????). The split of 32 into 24 and 8 could be varied and could depend on expected Training Set accuracy.  For accurate Training Sets, use, e.g., 12 HiBits (a very tight Hull), else use 4 HiBits only (a very loose Hull). When might we judge that the Training Set is very approximate? - When there are few Training points. Remember, for example, that the main criticism of most cancer prediction systems is that they are based on too few expert opinions or experimental cases because each is very expensive to obtain (i.e., we usually settle for just a few training points). While we’re at it, a new algorithm, Oblique-Hull, might only place hull segments when there is a gap between a pair of classes (separate segment pairs for each class pair). Continue to include new unit vectors until each class pair has been separated. The gap placement can use the 1st k HiBits value that produces a gap, k=1,2… (1st k HiBit value between the min inner product for one class and the max inner product for the other class). It seems like Mohammad’s 2’s complement procedure works that way anyway???? (proceeding one bit slice at a time from the high side? Or is it the low side?), so we can continually check for a gap and early exit as soon as one appears???? So for pair of Training Classes, we might use the unit vector between class means, then project the two classes using k HiBits only, k=1,2… (i.e., until a gap appears between the k HiBit min of the origin (of unit vector) class and the k HiBit max (plus 1) of the destination class …

  2. Bipart G11: Inv(12345) rec Stk(ABCDE) cliqueTrees NPZpTrst=5 L=2 L=1 L=0 G11 Stock EBCTs I S 1 A 1 1 1 1 1 1 1 2 3 4 5 0 0 1 1 1 0 1 1 1 0 3 3 0 0 0 0 1 0 1 1 1 0 1 3 1 1 0 0 0 1 1 1 1 0 2 4 0 1 0 0 0 1 1 1 1 1 1 5 0 1 1 1 0 0 0 1 1 1 3 3 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 A A A A A B B B B B C C C C C D D D D D E E E E E 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 1 1 0 3 3 1 1 1 1 0 0 0 1 1 0 4 2 0 1 0 1 0 1 0 1 1 1 2 4 2 B New DSs: Traditional data structures 3 C EdgeTbl A B C D E Stock EBCTs I S Investor BCTs S I Stock BCTs I S 1 1 1 1 2 2 2 2 2 3 3 3 4 4 4 4 Inv EBGTs 4 D A B C D A B C D E C D E A C D E A B C D E A B C D E 1 2 3 4 5 1 2 3 4 5 1 0 1 1 0 1 1 0 1 0 3 3 1 1 0 0 0 1 1 1 1 0 2 4 1 0 0 0 0 1 1 0 1 0 1 3 1 0 0 0 0 1 1 1 1 0 1 4 0 1 0 0 0 1 1 1 1 1 1 5 1 1 1 1 0 1 1 0 0 0 4 2 0 1 0 0 0 1 1 0 0 0 1 2 0 1 0 0 0 1 1 1 1 1 1 5 0 0 1 1 0 1 1 1 1 0 2 4 0 1 1 1 0 0 0 1 1 1 3 3 0 0 1 0 0 0 0 1 1 1 1 3 0 0 1 0 0 1 1 1 1 0 1 4 0 0 0 1 0 1 1 1 1 0 1 4 0 1 0 1 0 1 0 1 1 1 2 4 0 0 0 1 0 1 0 1 1 1 1 4 0 1 1 1 0 3 1 1 0 1 0 3 1 1 1 1 0 4 1 1 0 0 0 2 1 1 1 1 0 4 5 E oa 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 Graph oa A B C D E A B C D E 1 2 3 4 5 1 2 3 4 5 1 1 0 0 0 1 1 1 1 0 =C a MaxClique.Then 1 of must be a BC, say Expanding it gives C. Thus, for Bipartite Graphs, every MaxClique is an EBCT. =C a MaxClique.Then 1 of must be a BC, say Expanding it must give C. Thus, for Tripartite Graphs, every MaxClique is an EBCT. 1 1 1 1 1 0 2 1 1 1 1 1 3 0 0 1 1 1 4 1 0 1 1 1 5 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 A B C D E Adj Matrix 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 A B C D E A B C D E A B C D E A B C D E A B C D E 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 1 1 0 0 1 2 2 1 1 1 1 1 0 0 1 0 0 3 1 1 DI StockBaseCliqueTrees D I S H1: On Day() I(123) recommend S(ABC) Actually it is not true that 1 must be a BC, since there could be a different expansion for each of those 4, intersecting in C. In that case, we get each of those different expansions as an EBCT, but then the other operator will give us C (we will AND those expansions yielding the core leaf but OR the singletons giving the correct Part of C.             1 0 0 1 1 0 1 0 1 1 2 2 1 0 0 1 0 0 1 0 1 1 1 2 1 0 0 1 0 0 1 0 1 1 1 2 1 0 0 1 1 0 1 0 1 1 2 2 1 0 0 1 1 0 1 0 1 1 2 2 1 0 0 1 1 0 1 0 1 1 2 2 1 0 0 1 0 0 1 0 1 1 1 2 1 0 0 0 1 0 1 1 1 1 1 3 1 0 1 0 1 0 1 1 1 2 1 3 1 0 1 0 1 0 1 1 1 2 1 3 1 0 0 0 1 0 1 1 1 1 1 3 1 0 0 0 1 0 1 1 1 1 1 3 1 0 1 0 1 0 1 1 1 2 1 3 1 0 1 0 1 0 1 1 1 2 1 3 1 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 0 1 2 1 1 1 0 1 0 1 1 0 0 1 2 2 1 1 0 0 0 0 1 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 3 1 1 0 0 1 1 1 0 0 1 1 3 1 1 0 0 1 1 1 0 0 1 1 3 1 1 1 1 1 0 0 1 0 0 3 1 1 0 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 0 3 1 1 0 1 0 1 0 1 1 0 0 1 2 1 0 1 1 1 0 1 1 0 0 2 2 1 0 1 1 1 0 1 1 0 0 2 2 1 0 1 0 1 0 0 1 0 0 1 1 1 1 1 1 0 1 0 0 1 1 3 1 2 0 1 0 0 1 0 0 1 1 1 1 2 1 1 1 0 1 0 0 1 1 3 1 2 0 1 0 0 1 0 0 1 1 1 1 2 1 1 1 0 1 0 0 1 1 3 1 2 0 1 0 0 1 0 0 1 1 1 1 2 1 1 1 0 1 0 0 1 1 3 1 2 0 1 0 0 0 1 1 1 0 1 1 2 0 1 0 0 0 1 1 1 0 1 1 2 0 1 0 0 0 1 1 1 0 1 1 2 0 1 0 0 0 1 1 1 0 1 1 2 0 1 0 0 0 1 1 1 0 1 1 2 0 1 0 0 0 1 1 1 0 1 1 2 0 1 0 0 0 1 1 1 0 1 1 2 0 0 1 1 1 1 1 0 0 1 3 1 0 0 1 1 1 1 1 0 0 1 3 1 0 0 1 1 1 1 1 0 0 1 3 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 3 1 0 1 0 1 0 1 1 1 2 1 3 1 0 1 0 1 0 1 1 1 2 1 3 0 0 1 0 1 0 1 1 1 1 1 3 0 0 1 0 1 0 1 1 1 1 1 3 0 0 1 0 1 1 1 0 1 1 2 2 0 0 1 0 0 1 1 0 1 1 1 2 0 0 1 0 1 1 1 0 1 1 2 2 0 0 1 0 1 1 1 0 1 1 2 2 0 0 1 0 0 1 1 0 1 1 1 2 0 0 1 0 1 1 1 0 1 1 2 2 0 0 1 0 0 1 1 0 1 1 1 2 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 EdgeMap 1 2 3 1 2 3 DI StockBaseCliqueTrees D I S 1 2 3 1 2 3 A B C A B C A B C A B C aoa oaa       Thm: Every Maximal Clique is an Expanded Base Clique. I.e., C is a Maximal Clique iff C is an Expanded Base Clique I.e., MC(G)=EBC(G). Pf: Let be any MaxClique, C. Then some leaf expansion of each of … is a BCT. After we apply a..aoa..a with o in each but the last position, we will have an EBCT with the upper Parts of C and a leaf that covers the leaf of C. However, the leaf of that EBCT cannot strictly cover the leaf of C lest it be a MaxClique that strictly covers C. Thus, that EBCT=C H1 Stock EBCTs 1 2 3 1 2 3 A B C A B C NPZpTst=5 L=2 L=1 L=0 1 1 1 1 1 0 NPZ pTree (stride=3) L=3 L=2 L=1 L=0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 . . . 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 . . . 1 1 1 1 0 0 0 0 1 0 1 0 0 0 0 . . . 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 . . . 1 1 1 1 0 aoa oaa 0 1 0 1 0 1 1 0 0 0 . . . 1 1 1 1 0 1 1 1 1 1 1 1 1 1    1 1 1 1 0 4 1 1 1 1 1 5 0 0 1 1 1 3 1 0 1 1 1 4 1 0 1 2 1 1 1 3 0 0 1 1 1 0 0 1 0 1 1 2 1 1 0 2 1 0 0 1 1 1 1 3 1 0 1 2 1 2 3 A B C

  3. Vertical Graph Analyticsusing the Edge pTree (E) and the multi-Level PathPtree (PP) PP(G), the Path Ptree of graph, G, (undirected unipartite graph - but most of this can be modified for directed and bipartite graphs also). We use se PP(G) to find diameter, shortest paths, communities (both degree and density based, including cliques, k-cores and k-plexes) and motifs. Some of these measurements and existence theorems are NP-complete or NP-hard. Many assume this means “They can’t be done!” That assumption is what we’re addressing. By modifying the basic data structure (from the traditional, ubiquitous, horizontal RECORD to the beautiful, vertical pTree) it becomes in harmony with modern computing hardware’s strengths and we can do important Big Data NP computations quickly. Notes: If one creates PP(G), lots of tasks become easy! We wil always use the new pop-count facility which produces 1-counts duirng ANDs/ORs for free (timewise). C is a clique iff all C level-1 counts are |VC|-1. In fact one can mine all cliques by analyzing counts. k-plex existence: C is a k-plex iff vC|Cv|  |VC|2–k2 k-plex inheritance: Every induced subgraph of a k-plex is a k-plex. All max k-plexes: Use |Cv| vC A k-plex is a maximal subgraph in which each vertex is adjacent to all other vertices of the subgraph except at most k of them. A k-core is a maximal subgraph in which each vertex is adjacent to at least k other vertices of the subgraph. There is a whole hierarchy of cores of different order. All max k-cores: Use |Cv| vC k-core inheritance:If  cover by induced k-cores, G is k-core. k-core existence:C is a k-core iff vC,|VC|  k. Clique Existence: When is an induced SG a clique? Edge Count existence thm (EC): |EC||PUC|=COMB(|VC|,2)|VC|!/((|VC|-2)!2!) SubGraph existence thm (SG): (VC,EC) is a k-clique iff every induced k-1 subgraph, (VD,ED) is a (k-1)-clique. A Clique Mining alg: finds all cliques in a graph. For Clique-Mining we can use an ARM-Apriori-like downward closure property: CLQkkCliqueSet, CCLQk+1Candidatek+1CliqueSet By SG, CCLQk+1= all s of CLQk-pairs having k-1 common vertices. Let CCCLQk+1 be a union of two k-cliques with k-1 common vertices. Let v,w be the kth vertices of the k-cliques, then CCLQk+1 iff (PE)(v,w)=1. (Just need to check a single bit in PE.) Inter-cluster density δext(C)=|edges(C,C’)|/(nc(n-nc)) Int/Ext degree of v∈C, kvint/wxt=# edges v to wC/C’ Internal degree of C, kCint = vC kvint Intra-cluster density δint(C)=|edges(C,C)|/(nc(nc−1)/2) External degree of C, kCext =vC kvext The proper tradeoff between large δint(C) and small δext(C) is goal of many community mining algorithms. A simple approach is to Maximize differences. Density Difference algorithm for Communities: δint(C)−δext(C) >Threshold? Degree Differencealgorithm: kCint – kCext> Threshold? Easy to compute w pTrees, even for Big Graphs. Graphs are employed ubiquitously for complex data

  4. E3 143 E41342 The PathPtree for G1, PP(G1) 1 0 0 0 Two-Level Stride=4, Edge pTrees Two-Level Str=4, Unique Edge pTrees 0 0 0 0 E3 1 L13 14 L13 31 L13 43 L13 41 E3 112 E3 111 E3 134 E3 14 E3 121 E3 143 E3 141 E3 241 L23 1 L13 24 E3 243 E3 431 L13 34 E3 314 E3 113 L23 4 E3 413 E3 341 L23 3 L13 13 E3 114 L23 2 E3 12 E3 241 E3 134 E3 314 E3 341 E3 144 E3 342 E3 413 E3 122 E3 143 E3 142 E3 431 E3 243 E3 123 E3 11 E3 131 E3 132 E3 124 E3 134 E3 13 E3 142 E3 133 E3 2 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 h=1 j=4 k=2 E3142=E2&M’4 E3 E3 E3 M’4 1 1 1 0 E3key E2 0 0 0 1 1 E3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1111 1112 1113 1114 1121 1122 1123 1124 1131 1132 1133 1134 1141 1142 1143 1144 1211 1212 1213 1214 1221 1222 1223 1224 1231 1232 1233 1234 1241 1242 1243 1244 1311 1312 1313 1314 1321 1322 1323 1324 1331 1332 1333 1334 1341 1342 1343 1344 1411 1412 1413 1414 1421 1422 1423 1424 1431 1432 1433 1434 1441 1442 1443 1444 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 pure0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2111 2112 2113 2114 2121 2122 2123 2124 2131 2132 2133 2134 2141 2142 2143 2144 2211 2212 2213 2214 2221 2222 2223 2224 2231 2232 2233 2234 2241 2242 2243 2244 2311 2312 2313 2314 2321 2322 2323 2324 2331 2332 2333 2334 2341 2342 2343 2344 2411 2412 2413 2414 2421 2422 2423 2424 2431 2432 2433 2434 2441 2442 2443 2444 3111 3112 3113 3114 3121 3122 3123 3124 3131 3132 3133 3134 3141 3142 3143 3144 3211 3212 3213 3214 3221 3222 3223 3224 3231 3232 3233 3234 3241 3242 3243 3244 3311 3312 3313 3314 3321 3322 3323 3324 3331 3332 3333 3334 3341 3342 3343 3344 3411 3412 3413 3414 3421 3422 3423 3424 3431 3432 3433 3434 3441 3442 3443 3444 4111 4112 4113 4114 4121 4122 4123 4124 4131 4132 4133 4134 4141 4142 4143 4144 4211 4212 4213 4214 4221 4222 4223 4224 4231 4232 4233 4234 4241 4242 4243 4244 4311 4312 4313 4314 4321 4322 4323 4324 4331 4332 4333 4334 4341 4342 4343 4344 4411 4412 4413 4414 4421 4422 4423 4424 4431 4432 4433 4434 4441 4442 4443 4444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 L1 1 1 1 0 L1 E 1 1 1 1 Edge Mask pTree h=2 j=4 ListE224={1,3} k=1 E3241=E1&M’4 Unique Edge Mask V1 L0 Vertex Masks 1 2 3 4 V2 L0 M’4 1 1 1 0 E1 0 0 1 1 Edges V1 V2 3 4 U 0 0 1 1_ 0 0 0 1_ 0 0 0 1_ 0 0 0 0 E 0 0 1 1_ 0 0 0 1_ 1 0 0 1_ 1 1 1 0 1 E1 0 0 1 1 U1 0 0 1 1 M1 1 0 0 0 1,1 1,2 1,3 1,4_ 2,1 2,2 2,3 2,4_ 3,1 3,2 3,3 3,4_ 4,1 4,2 4,3 4,4 2 1 3 E2 0 0 0 1 U2 0 0 0 1 M2 0 1 0 0 h=2 j=4 k=3 E3243=E3&M’4 1Lev EE 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 Str=4 3level EE11 0 0 0 0 Str=16 2Level EE1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 Graph Path: a Sequence of edges connecting a sequence of vertices which are distinct from each other except for the endpts ( other defs?). 4 1 1 1 M’4 1 1 1 0 E3 1 0 0 1 111 112 113 114 121 122 123 124 131 132 133 134 141 142 143 144 211 212 213 214 221 222 223 224 231 232 233 234 241 242 243 244 311 312 313 314 321 322 323 324 331 332 333 334 341 342 343 344 411 412 413 414 421 422 423 424 431 432 433 434 441 442 443 444 E3 1 0 0 1 U3 0 0 0 1 M3 0 0 1 0 EE12 0 0 0 0 E4 1 1 1 0 U4 0 0 0 0 M4 0 0 0 1 2paths = E2, 3paths = E3, etc. E3& 1 0 0 1 M’1= 0 1 1 1 EE13 0 0 0 1 EE13 0 0 0 1 E4 1 1 1 0 M’1 0 1 1 1 h=3 j=1 k=4 E3314=E4&M’1 E2key v1v2v3 kListEh, E2hk=Ek&M’h (other k, E2hk=0) EE14 0 1 1 0 E4 1 1 1 0 M’1= 0 1 1 1 EE14 0 1 1 0 For h=1, ListE1={3,4} EE2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 EE21 0 0 0 0 E1 0 0 1 1 M’4 1 1 1 0 h=3 j=4 k=1 E3341=E1&M’4 For h=1 k=3: EE13=E3&M’1 EE22 0 0 0 0 EE23 0 0 0 0 For h=1 k=4: EE14=E4&M’1 E2 0 0 0 1 M’4 1 1 1 0 h=3 j=4 k=2 E3342=E2&M’4 EE24 1 0 1 0 M’2 1 0 1 1 EE24 1 0 1 0 E4 1 1 1 0 EE31 0 0 0 1 EE3 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 For h=2, ListE2={4} E1& 0 0 1 1 M’3= 1 1 0 1 EE31 0 0 0 1 EE32 0 0 0 0 E3 1 0 0 1 M’1 0 1 1 1 h=4 j=1 k=3 E3413=E3&M’1 E4& 1 1 1 0 M’3= 1 1 0 1 EE34 1 1 0 0 EE33 0 0 0 0 For h=2 k=4: EE24=E4&M’2 For h=3, ListE3={1,4} EE34 1 1 0 0 E1 0 0 1 1 M’3 1 1 0 1 h=4 j=3 k=1 E3431=E1&M’3 For h=3 k=1: EE31=E1&M’3 E1& 0 0 1 1 M’4= 1 1 1 0 EE41 0 0 1 0 EE41 0 0 1 0 EE4 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 kListE2hj, E3hjk=Ek&M’j. E4 1 1 1 0 M’3 1 1 0 1 h=1 j=3 ListE213={4} k=4 E3134=E4&M’3 EE42 0 0 0 0 E2& 0 0 0 1 M’4= 1 1 1 0 h=1 j=4 k=3 E3143=E3&M’4 EE42 0 0 0 0 pure0 M’4 1 1 1 0 E3 1 0 0 1 For h=3 k=4: EE34=E4&M’3 EE43 1 0 0 0 kListE3hij, E4hijk = Ek & M’j & M’i ListE3143={1} For h=4, ListE4={1,2,3} EE44 0 0 0 0 E3& 1 0 0 1 M’4= 1 1 1 0 EE43 1 0 0 0 For h=4 k=1: EE41=E1&M’4 Level=2 (These are exactly the Level=1 of E2) For h=4 k=2: EE42=E2&M’4 E42413 E43142 E42431 M’1 0 1 1 1 M’3 1 1 0 1 M’1 0 1 1 1 E3 0 0 0 1 E2 0 0 0 1 E1 0 0 1 1 M’4 1 1 1 0 M’4 1 1 1 0 M’4 1 1 1 0 M’3 1 1 0 1 E2 0 0 0 1 M’4 1 1 1 0 ListE3134={1,2} h=1 i=3 j=4 k=2 Level=3(So E2 is the upper 3 levels of E3) 0 0 0 0 0 0 0 0 0 0 0 0 ListE3243={1} h=2 i=4 j=3 k=1 ListE3314={2,3} h=3 i=1 j=4 k=2 ListE3241={3} h=2 i=4 j=1 k=3 Level=2 For h=4 k=3: EE43=E3&M’4 1 1 1 1 ListE3341={3} 1 1 1 1 ListE3413={4} Level=1 (These are exactly the Level=0’s of E2) ListE3431={4} EE=E2: 3 Level Stri=4 pTrees for Path Len=2 (2edges, 3vertices, unique except for endpts) Level=1=just E1,E2,E3,E4 with pure0 bits turned off. E1 0 0 1 1 E2 0 0 0 1 E3 1 0 0 1 E4 1 0 bit turned off 1 0 There are no 5vertex (4edge) paths. Creation stops. The Stride=|V|, Levels=Diam PathPtree (PP): E E2 E3 : Elongest_path Level=0 (We just computed these) Level=0 EE13 0 0 0 1 EE14 0 1 1 0 EE24 1 0 1 0 EE34 1 1 0 0 EE31 0 0 0 1 EE41 0 0 1 0 EE43 1 0 0 0

  5. G1 Use PP to get other pTrees for Shortest Paths, Diameter, Unique Paths, Cycles, Unique Cycles, Unique Acyclic Cycles Apply to more Graphs. Revising PP when edges are added. 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 1 0 1 1 1 2 1 0 0 1 3 1 0 0 1 4 1 1 1 0 2 2 1 1 1 G11 Unique PathPtree (UPP): Top-bottom, left-right, eliminate paths ending w the starting vertex after all pTrees with that starter have been included. DIAMETER(G1)? PP(G1) SP(G1)? PP(G1) 1 3 0 0 0 1 1 4 0 1 1 0 2 4 1 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 1 2 0 0 0 1 1 3 0 0 0 1 1 4 0 1 1 0 2 1 0 0 1 1 2 4 1 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 4 2 0 0 0 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 3 0 0 0 1 4 1 1 1 0 4 0 1 1 0 4 0 0 1 0 4 0 0 0 0 No 3 3 3 4 4 4 1 3 4 1 0 0 0 1 4 3 1 0 0 0 3 4 2 0 0 0 0 3 1 4 0 1 1 0 1 3 4 1 1 0 0 1 4 2 0 0 0 0 2 4 1 0 0 1 0 2 4 3 1 0 0 0 3 4 1 0 0 1 0 4 1 3 0 0 0 1 4 3 1 0 0 0 1 1 3 0 0 0 1 1 4 0 1 1 0 2 4 1 0 1 0 2 4 0 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 3 4 0 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 3 4 0 0 0 0 4 1 0 0 0 0 4 3 0 0 0 0 1 2 4 0 0 1 0 1 3 4 1 1 0 0 1 4 2 1 0 0 0  So SP1,2=132 Add path (12). Have to rebuild entire PP? no kListEh, E2hk=Ek kill h Diam1=max{111212}=2 Diam2=3 Diam4=3 Diam3=3 Diam5=3 Diam6=2 Diam7=3 1 4 3 1 0 0 0 3 4 2 0 0 0 0 3 1 4 0 1 1 0 1 3 4 1 1 0 0 1 4 2 0 0 0 0 2 4 1 0 0 1 0 2 4 3 0 0 0 0 2 4 3 1 0 0 0 3 4 1 0 0 1 0 4 1 3 0 0 0 1 Cycles List (CL): 1341 1431 3143 3413 CP cycles only. includes all Nonredundant Cycles. kListE2hj, E3hjk=Ek kill j 4 3 1 0 0 0 1 3 1 4 0 0 1 0 4 3 1 0 0 0 0 1 0 1 1 1 0 1 0 2 1 0 1 1 0 0 0 3 1 1 0 1 0 0 0 4 1 1 1 0 0 0 0 5 0 0 0 0 0 1 0 6 1 0 0 0 1 0 1 7 0 0 0 0 0 1 0 G2 PP(G2) So, DiamG2=maxkV(Diamk)=3 kListE3hij E4hijk=Ek kill i,j Diamk=minhkPathLen(h,k). k, record the Ek depth level of the 1st occurrence of vertex h, hk. DiamG=maxkGDiamk. Eliminate 31 and 41 1 2 0 0 1 1 0 0 0 1 3 0 1 0 1 0 0 0 1 4 0 1 1 0 0 0 0 1 6 0 0 0 0 1 0 1 2 1 0 0 1 1 0 1 0 2 3 1 0 0 1 0 0 0 2 4 1 0 1 0 0 0 0 3 1 0 1 0 1 0 1 0 3 2 1 0 0 1 0 0 0 3 4 1 1 0 0 0 0 0 4 1 0 1 1 0 0 1 0 4 2 1 0 1 0 0 0 0 4 3 1 1 0 0 0 0 0 5 6 1 0 0 0 0 0 1 6 1 0 1 1 1 0 0 0 7 6 1 0 0 0 1 0 0 Eliminate 42 G3 Let’s see what happens if we add an edge to G2 Eliminate Diam1=max{fo12 fo13 fo14}. What is the first occurrence 12, fo12? The PPdepth from E1 where 2 first appears is depth= 2 = fo12 So Diam1=max{2 1 1}= 2 Can UCP be used to mine all cliques? since a cliques must be cycles at each level. (In this example, there is only 1 level to check since there can be no 2cycles (2edges 3vertices.). PP Clique Mine Algs: Every clique is made up entirely of cycles at all levels. Every 3cycle is clique  4cycle, abcda check edges ac bd. Also check each acyclic 4path abcde for edges ac ad ae bd de ce. If any missing, eliminate branch, else look for acyclic 4path, 5paths … in its pTree branch, etc. Extend to Path pTree k-plex(k-core) mining algorithm? 1 0 0 1 1 2 0 0 0 1 3 0 0 0 1 1 4 3 1 1 0 0 0 0 0 3 1 2 0 0 1 1 0 0 0 1 3 2 1 0 0 1 0 0 0 1 3 4 1 1 0 0 0 0 0 1 4 2 1 0 1 0 0 0 0 1 2 3 1 0 0 1 0 0 0 1 2 4 1 0 1 0 0 0 0 2 1 4 0 1 1 0 0 0 0 2 1 6 0 0 0 0 1 0 1 2 4 1 0 1 1 0 0 1 0 3 1 4 0 1 1 0 0 0 0 2 1 3 0 1 0 1 0 0 0 2 3 1 0 1 0 1 0 1 0 2 3 4 1 1 0 0 0 0 0 2 4 3 1 1 0 0 0 0 0 3 1 6 0 0 0 0 1 0 1 3 2 1 0 0 1 1 0 1 0 3 2 4 1 0 1 0 0 0 0 3 4 1 0 1 1 0 0 1 0 3 4 2 1 0 1 0 0 0 0 4 1 2 0 0 1 1 0 0 0 4 1 3 0 1 0 1 0 0 0 4 1 6 0 0 0 0 1 0 1 4 2 1 0 0 1 1 0 1 0 4 2 3 1 0 0 1 0 0 0 4 3 1 0 1 0 1 0 1 0 4 3 2 1 0 0 1 0 0 0 5 6 1 0 1 1 1 0 0 0 6 1 2 0 0 1 1 0 0 0 6 1 3 0 1 0 1 0 0 0 6 1 4 0 1 1 0 0 0 0 7 6 1 0 1 1 1 0 0 0 1 3 0 0 0 1 2 4 0 0 1 0 1 4 0 1 1 0 3 1 0 0 0 1 Diam2=max{fo21 fo23 fo24}=max{2 2 1}= 2 Diam3=max{fo31 fo32 fo34}=max{1 2 1}= 2 PP update alg? Copy 2paths, add new ones… 1 3 4 1 1 0 0 3 1 4 0 0 1 0 4 1 3 0 0 0 1 1 4 3 1 0 0 0 2 4 1 0 0 1 0 3 4 1 0 0 1 0 Diam4=max{fo41 fo42 fo43}=max{1 1 1}= 1 4 1 3 2 0 0 0 1 0 0 0 2 3 1 6 0 0 0 0 1 0 1 1 2 3 4 1 0 0 0 0 0 0 1 2 4 3 1 0 0 0 0 0 0 1 3 2 4 1 0 0 0 0 0 0 1 3 4 2 1 0 0 0 0 0 0 1 4 2 3 1 0 0 0 0 0 0 1 4 3 2 1 0 0 0 0 0 0 2 1 3 4 0 1 0 0 0 0 0 2 1 4 3 0 1 0 0 0 0 0 2 3 1 4 0 1 0 0 0 0 0 2 3 4 1 0 1 0 0 0 1 0 2 4 1 3 0 1 0 0 0 0 0 2 4 1 6 0 0 0 0 1 0 1 3 1 2 4 0 0 1 0 0 0 0 3 1 4 2 0 0 1 0 0 0 0 3 2 1 6 0 0 0 0 1 0 1 3 4 1 2 0 0 1 0 0 0 0 3 4 1 6 0 0 0 0 1 0 1 3 4 2 1 0 0 1 0 0 1 0 4 1 2 3 0 0 0 1 0 0 0 4 2 1 3 0 0 0 1 0 0 0 4 2 1 6 0 0 0 0 1 0 1 4 2 3 1 0 0 0 1 0 1 0 4 3 1 2 0 0 0 1 0 0 0 4 3 1 6 0 0 0 0 1 0 1 4 3 2 1 0 0 0 1 0 1 0 5 6 1 2 0 0 1 1 0 0 0 5 6 1 3 0 1 0 1 0 0 0 5 6 1 4 0 1 1 0 0 0 0 6 1 2 3 0 0 0 1 0 0 0 6 1 2 4 0 0 1 0 0 0 0 6 1 3 2 0 0 0 1 0 0 0 6 1 3 4 0 1 0 0 0 0 0 6 1 4 2 0 0 1 0 0 0 0 6 1 4 3 0 1 0 0 0 0 0 7 6 1 2 0 0 1 1 0 0 0 7 6 1 3 0 1 0 1 0 0 0 7 6 1 4 0 1 1 0 0 0 0 3 2 4 1 0 0 1 0 0 1 0 2 4 3 1 0 1 0 0 0 1 0 5 7 0 0 0 0 0 1 0 7 5 0 0 0 0 0 1 0 6 6 DiamG1=maxkV(Diamk) = 2 5 5 To find a Shortest Path from h to k? (path of length minhkPathLengthh,k), go down PP from Ek until h first appears. Eg, SP(1,2)? 7 7 CP UCP 7 5 6 1 0 0 0 1 0 1 1 3 4 1 0 0 0 5 7 6 1 0 0 0 1 0 0 3 1 4 0 0 1 0 1 4 3 1 0 0 0 3 4 1 0 0 1 0 SP in G2? SP(7,2)? SP(1,5)? 1 0 1 1 1 0 1 0 2 1 0 1 1 0 0 0 3 1 1 0 1 0 0 0 4 1 1 1 0 0 0 0 5 0 0 0 0 0 1 1 6 1 0 0 0 1 0 1 7 0 0 0 0 1 1 0 2 2 no 1 1 2 1 6 5 0 0 0 0 0 0 1 2 1 6 7 0 0 0 0 1 0 0 3 1 6 5 0 0 0 0 0 0 1 3 1 6 7 0 0 0 0 1 0 0 4 1 6 5 0 0 0 0 0 0 1 4 1 6 7 0 0 0 0 1 0 0 no 1 1 clockwise counterclockwise 3 3 4 4 2 4 3 1 0 1 0 0 0 1 0 1 2 0 0 1 1 0 0 0 1 3 0 1 0 1 0 0 0 1 4 0 1 1 0 0 0 0 1 6 0 0 0 0 1 0 1 2 1 0 0 1 1 0 1 0 2 3 1 0 0 1 0 0 0 2 4 1 0 1 0 0 0 0 3 1 0 1 0 1 0 1 0 3 2 1 0 0 1 0 0 0 3 4 1 1 0 0 0 0 0 4 1 0 1 1 0 0 1 0 4 2 1 0 1 0 0 0 0 4 3 1 1 0 0 0 0 0 5 6 1 0 0 0 0 0 1 6 1 0 1 1 1 0 0 0 7 6 1 0 0 0 1 0 0 no y, SP15=165. 4 3 2 1 6 0 0 0 0 1 0 1 5 6 1 2 3 0 0 0 1 0 0 0 5 6 1 2 4 0 0 1 0 0 0 0 2 3 1 6 5 0 0 0 0 0 0 1 2 3 1 6 7 0 0 0 0 1 0 0 2 3 4 1 6 0 0 0 0 1 0 1 2 4 1 6 7 0 0 0 0 1 0 0 2 4 3 1 6 0 0 0 0 1 0 1 3 2 4 1 6 0 0 0 0 1 0 1 3 2 1 6 5 0 0 0 0 0 0 1 3 2 1 6 7 0 0 0 0 1 0 0 3 4 1 6 5 0 0 0 0 0 0 1 3 4 1 6 7 0 0 0 0 1 0 0 3 4 2 1 6 0 0 0 0 1 0 1 4 2 1 6 5 0 0 0 0 0 0 1 4 2 1 6 7 0 0 0 0 1 0 0 4 2 3 1 6 0 0 0 0 1 0 1 4 3 1 6 5 0 0 0 0 0 0 1 4 3 1 6 7 0 0 0 0 1 0 0 5 6 1 4 3 0 1 0 0 0 0 0 5 6 1 3 2 0 0 0 1 0 0 0 5 6 1 3 4 0 1 0 0 0 0 0 2 4 1 6 5 0 0 0 0 0 0 1 5 6 1 4 2 0 0 1 0 0 0 0 7 6 1 2 3 0 0 0 1 0 0 0 7 6 1 2 4 0 0 1 0 0 0 0 7 6 1 3 2 0 0 0 1 0 0 0 7 6 1 3 4 0 1 0 0 0 0 0 7 6 1 4 2 0 0 1 0 0 0 0 7 6 1 4 3 0 1 0 0 0 0 0 1 4 3 1 1 0 0 0 0 0 3 1 2 0 0 1 1 0 0 0 1 3 2 1 0 0 1 0 0 0 1 3 4 1 1 0 0 0 0 0 1 4 2 1 0 1 0 0 0 0 1 2 3 1 0 0 1 0 0 0 1 2 4 1 0 1 0 0 0 0 2 1 4 0 1 1 0 0 0 0 2 1 6 0 0 0 0 1 0 1 2 4 1 0 1 1 0 0 1 0 3 1 4 0 1 1 0 0 0 0 2 1 3 0 1 0 1 0 0 0 2 3 1 0 1 0 1 0 1 0 2 3 4 1 1 0 0 0 0 0 2 4 3 1 1 0 0 0 0 0 3 1 6 0 0 0 0 1 0 1 3 2 1 0 0 1 1 0 1 0 3 2 4 1 0 1 0 0 0 0 3 4 1 0 1 1 0 0 1 0 3 4 2 1 0 1 0 0 0 0 4 1 2 0 0 1 1 0 0 0 4 1 3 0 1 0 1 0 0 0 4 1 6 0 0 0 0 1 0 1 4 2 1 0 0 1 1 0 1 0 4 2 3 1 0 0 1 0 0 0 4 3 1 0 1 0 1 0 1 0 4 3 2 1 0 0 1 0 0 0 5 6 1 0 1 1 1 0 0 0 6 1 2 0 0 1 1 0 0 0 6 1 3 0 1 0 1 0 0 0 6 1 4 0 1 1 0 0 0 0 7 6 1 0 1 1 1 0 0 0 4 4 3 3 y, SP72 =7612 2 3 4 1 6 5 0 0 0 0 0 0 1 2 3 4 1 6 7 0 0 0 0 1 0 0 2 4 3 1 6 5 0 0 0 0 0 0 1 2 4 3 1 6 7 0 0 0 0 1 0 0 4 2 3 1 6 5 0 0 0 0 0 0 1 4 2 3 1 6 7 0 0 0 0 1 0 0 4 3 2 1 6 5 0 0 0 0 0 0 1 4 3 2 1 6 7 0 0 0 0 1 0 0 3 2 4 1 6 5 0 0 0 0 0 0 1 3 2 4 1 6 7 0 0 0 0 1 0 0 3 4 3 1 6 5 0 0 0 0 0 0 1 3 4 3 1 6 7 0 0 0 0 1 0 0 3 2 4 1 6 5 0 0 0 0 0 0 1 3 2 4 1 6 7 0 0 0 0 1 0 0 3 4 3 1 6 5 0 0 0 0 0 0 1 3 4 3 1 6 7 0 0 0 0 1 0 0 Next, complete added levels. (5paths exist now whereas they didn’t before. Also 6paths) 4 1 3 2 0 0 0 1 0 0 0 2 3 1 6 0 0 0 0 1 0 1 1 2 3 4 1 0 0 0 0 0 0 1 2 4 3 1 0 0 0 0 0 0 1 3 2 4 1 0 0 0 0 0 0 1 3 4 2 1 0 0 0 0 0 0 1 4 2 3 1 0 0 0 0 0 0 1 4 3 2 1 0 0 0 0 0 0 2 1 3 4 0 1 0 0 0 0 0 2 1 4 3 0 1 0 0 0 0 0 2 3 1 4 0 1 0 0 0 0 0 2 3 4 1 0 1 0 0 0 1 0 2 4 1 3 0 1 0 0 0 0 0 2 4 1 6 0 0 0 0 1 0 1 3 1 2 4 0 0 1 0 0 0 0 3 1 4 2 0 0 1 0 0 0 0 3 2 1 6 0 0 0 0 1 0 1 3 4 1 2 0 0 1 0 0 0 0 3 4 1 6 0 0 0 0 1 0 1 3 4 2 1 0 0 1 0 0 1 0 4 1 2 3 0 0 0 1 0 0 0 4 2 1 3 0 0 0 1 0 0 0 4 2 1 6 0 0 0 0 1 0 1 4 2 3 1 0 0 0 1 0 1 0 4 3 1 2 0 0 0 1 0 0 0 4 3 1 6 0 0 0 0 1 0 1 4 3 2 1 0 0 0 1 0 1 0 5 6 1 2 0 0 1 1 0 0 0 5 6 1 3 0 1 0 1 0 0 0 5 6 1 4 0 1 1 0 0 0 0 6 1 2 3 0 0 0 1 0 0 0 6 1 2 4 0 0 1 0 0 0 0 6 1 3 2 0 0 0 1 0 0 0 6 1 3 4 0 1 0 0 0 0 0 6 1 4 2 0 0 1 0 0 0 0 6 1 4 3 0 1 0 0 0 0 0 7 6 1 2 0 0 1 1 0 0 0 7 6 1 3 0 1 0 1 0 0 0 7 6 1 4 0 1 1 0 0 0 0 3 2 4 1 0 0 1 0 0 1 0

  6. 2Lev Str=4 SPTG1 (initially E) G1 Retaining the Shortest Path So Far structure? SPT(G)k (with k turned on) is a mask (where >0 means “yes”) for connectivity comp, COMP(G)k, containing the vertex, vk. For a bitmap of COMPk bit-slicing SPT (SPTk,h ... SPTk,0 k=1…|V|), then COMPk ORj=h..0SPTk,h. The SPT structure may be more useful expressed as separate “categorical” bitmaps for each Shortest Path Length (SPk,h h=1..H. We keep a mask of Shortest Paths so far, SPSFk vertex, k. With each new SP bitmap, SPB, SPSFkSPSFk| SPB and SPk,h+1 SPB & SPSFk. 2 1 APPT 1 1 1 1 E 1 1 1 1 SPT 1 1 1 1 APPT PPT 1 1 1 1 SPTG1, initially E1=SP1,1=SPSF1 E2=SP2,1=SPSF2 E3=SP3,1=SPSF3 E4=SP4,1=SPSF4 1 0 0 1 1 1 0 0 1 1 2 0 0 0 1 2 0 0 0 1 3 1 0 0 1 3 1 0 0 1 4 1 1 1 0 4 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 2 0 0 0 1 2 1 0 0 0 1 3 1 1 0 0 1 4 1 1 1 1 0 3 1 0 0 1 4 1 1 1 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 3 4 E one-level EG1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 key 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 1 3 0 0 0 1 1 4 0 1 1 0 2 4 1 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 SPT gives Connectivity Partition. For Maximal Cliques (go across SPk,1look in subsets of those k’s for commonality); Cliques are 0-plexes. Each SPk,1 masks a 1plex. Each SPk,1&SPk,2 masks a 2-plex (=SPSFk,2?) So if we save each SPSF instead of overwriting, we will have the k-plex masks without any further work??), etc. 1 3 4 1 1 0 0 1 4 3 1 0 0 0 2 4 1 0 0 1 0 2 4 3 1 0 0 0 3 1 4 0 1 1 0 4 3 1 0 0 0 1 3 4 1 0 0 1 0 4 1 3 0 0 0 1 1 3 0 0 0 1 1 4 0 1 1 0 2 4 1 0 1 0 3 1 0 0 0 1 3 4 1 1 0 0 4 1 0 0 1 0 4 3 1 0 0 0 1 0 2 1 1 2 2 0 2 1 3 1 2 0 1 For Big Graphs, could stop here (e.g., Friends has ~1B vertices but a diameter of 4, so would only need to build PT 4-hop paths) and possible expressed as a tree of lists rather than a tree of bitmaps. Also, for sparse BigGraphs, E could be leveled further. 1 2 0 1 0 0 3 2 0 1 0 0 SPSFk 1 3 4 0 1 0 0 2 4 1 0 0 1 0 2 4 3 1 0 0 0 3 1 4 0 1 0 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 All are 3 hop cycles. Each has 3 start pts and 2 directions. Each repeats 6 times. 6/6=1 3hop cycles (1341) 2 2 1 0 1 0 CLG1 1341 1431 3143 3413 4134 4314

  7. More PP, UPP, SPT, CL, UCL… Form UPP(G2) from PP(G2): Top-bottom, Left-right: After all pTrees with a given start vertex have been included, eliminate paths ending with that start vertex. 1 0 1 1 1 0 1 0 2 1 0 1 1 0 0 0 2 0 0 1 1 0 0 0 3 1 1 0 1 0 0 0 3 0 1 0 1 0 0 0 3 0 0 0 1 0 0 0 4 1 1 1 0 0 0 0 4 0 1 1 0 0 0 0 4 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 6 1 0 0 0 1 0 1 6 0 0 0 0 0 0 1 6 0 0 0 0 1 0 1 7 0 0 0 0 0 1 0 7 0 0 0 0 0 0 0 Cycles List CL(G2) (only 3,4cycles.  no 1,2 cycles) 1 4 3 1 3 1 2 3 1 3 2 1 1 3 4 1 1 4 2 1 1 2 3 1 1 2 4 1 2 1 4 2 2 4 1 2 3 1 4 3 2 1 3 2 2 3 1 2 2 3 4 2 2 4 3 2 3 2 1 3 3 2 4 3 3 4 1 3 3 4 2 3 4 1 2 4 4 1 3 4 4 2 1 4 4 2 3 4 4 3 1 4 4 3 2 4 3 1 0 0 0 1 0 1 0 3 4 0 0 0 0 0 0 0 4 1 0 0 1 0 0 1 0 6 1 0 0 1 1 0 0 0 2 1 0 0 1 1 0 1 0 2 3 1 0 0 1 0 0 0 2 3 0 0 0 1 0 0 0 2 4 1 0 1 0 0 0 0 2 4 0 0 1 0 0 0 0 3 1 0 1 0 1 0 1 0 3 2 1 0 0 1 0 0 0 3 2 0 0 0 1 0 0 0 3 4 1 1 0 0 0 0 0 3 4 0 1 0 0 0 0 0 4 1 0 0 0 0 0 1 0 4 1 0 1 1 0 0 1 0 4 2 1 0 1 0 0 0 0 4 2 0 0 1 0 0 0 0 4 3 1 1 0 0 0 0 0 4 3 0 1 0 0 0 0 0 5 6 1 0 0 0 0 0 1 5 6 0 0 0 0 0 0 1 6 1 0 0 0 0 0 0 0 6 1 0 0 0 1 0 0 0 6 1 0 1 1 1 0 0 0 7 6 1 0 0 0 1 0 0 7 6 0 0 0 0 1 0 0 4 2 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 0 7 6 0 0 0 0 0 0 0 1 2 0 0 1 1 0 0 0 1 2 0 0 1 1 0 0 0 1 3 0 1 0 1 0 0 0 1 3 0 1 0 1 0 0 0 1 4 0 1 1 0 0 0 0 1 4 0 1 1 0 0 0 0 1 6 0 0 0 0 1 0 1 1 6 0 0 0 0 1 0 1 4 1 3 2 4 1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 1 3 4 2 1 1 4 2 3 1 1 4 3 2 1 2 1 3 4 2 2 1 4 3 2 2 3 1 4 2 2 3 4 1 2 2 4 1 3 2 3 1 2 4 3 3 1 4 2 3 3 4 1 2 3 3 4 2 1 3 4 1 2 3 4 4 2 1 3 4 4 2 3 1 4 4 3 1 2 4 4 3 2 1 4 3 2 4 1 3 2 4 3 1 2 1 4 3 1 1 0 0 0 0 0 3 1 2 0 0 1 1 0 0 0 1 3 2 1 0 0 1 0 0 0 1 3 4 1 1 0 0 0 0 0 1 4 2 1 0 1 0 0 0 0 1 2 3 1 0 0 1 0 0 0 1 2 4 1 0 1 0 0 0 0 2 1 4 0 1 1 0 0 0 0 2 1 6 0 0 0 0 1 0 1 2 4 1 0 1 1 0 0 1 0 3 1 4 0 1 1 0 0 0 0 2 1 3 0 1 0 1 0 0 0 2 3 1 0 1 0 1 0 1 0 2 3 4 1 1 0 0 0 0 0 2 4 3 1 1 0 0 0 0 0 3 1 4 0 0 1 0 0 0 0 3 1 6 0 0 0 0 1 0 1 3 2 1 0 0 1 1 0 1 0 3 2 4 1 0 1 0 0 0 0 3 4 1 0 1 1 0 0 1 0 3 4 2 1 0 1 0 0 0 0 2 3 4 0 1 0 0 0 0 0 2 4 3 0 1 0 0 0 0 0 3 4 1 0 0 1 0 0 1 0 3 2 4 0 0 1 0 0 0 0 3 4 2 0 0 1 0 0 0 0 4 1 2 0 0 1 1 0 0 0 4 1 3 0 1 0 1 0 0 0 4 1 6 0 0 0 0 1 0 1 4 2 1 0 0 1 1 0 1 0 4 2 3 1 0 0 1 0 0 0 4 3 1 0 1 0 1 0 1 0 4 3 2 1 0 0 1 0 0 0 5 6 1 0 1 1 1 0 0 0 6 1 2 0 0 1 1 0 0 0 6 1 3 0 1 0 1 0 0 0 6 1 4 0 1 1 0 0 0 0 7 6 1 0 1 1 1 0 0 0 4 1 2 0 0 0 1 0 0 0 4 1 3 0 0 0 1 0 0 0 4 2 1 0 0 0 1 0 1 0 4 3 1 0 0 0 1 0 1 0 5 6 1 0 0 1 1 0 0 0 6 1 2 0 0 0 1 0 0 0 6 1 3 0 0 0 1 0 0 0 6 1 4 0 0 1 0 0 0 0 7 6 1 0 0 1 1 0 0 0 5 6 1 0 0 0 1 0 0 0 6 1 4 0 0 0 0 0 0 0 7 6 1 0 0 0 1 0 0 0 4 2 3 0 0 0 1 0 0 0 4 3 2 0 0 0 1 0 0 0 6 1 2 0 0 0 0 0 0 0 6 1 3 0 0 0 0 0 0 0 5 6 1 0 0 0 0 0 0 0 7 6 1 0 0 0 0 0 0 0 G2 PP(G2) 1 0 1 1 1 0 1 0 2 0 0 1 1 0 0 0 3 0 0 0 1 0 0 0 5 0 0 0 0 0 1 0 6 0 0 0 0 0 0 1 2 1 0 0 1 1 0 1 0 2 3 1 0 0 1 0 0 0 2 3 0 0 0 1 0 0 0 2 4 1 0 1 0 0 0 0 2 4 0 0 1 0 0 0 0 3 1 0 0 0 1 0 1 0 5 6 0 0 0 0 0 0 1 3 2 0 0 0 1 0 0 0 4 1 0 0 0 0 0 1 0 Unique PathPtree for G2, UPP(G2) 1 4 3 0 1 0 0 0 0 0 3 1 2 0 0 0 1 0 0 0 1 3 2 0 0 0 1 0 0 0 1 3 4 0 1 0 0 0 0 0 1 4 2 0 0 1 0 0 0 0 1 2 3 0 0 0 1 0 0 0 1 2 4 0 0 1 0 0 0 0 2 1 4 0 0 1 0 0 0 0 2 1 6 0 0 0 0 1 0 1 2 4 1 0 0 1 0 0 1 0 3 1 4 0 1 0 0 0 0 0 2 1 3 0 0 0 1 0 0 0 2 3 1 0 0 0 1 0 1 0 2 3 4 1 0 0 0 0 0 0 2 4 3 1 0 0 0 0 0 0 3 1 6 0 0 0 0 1 0 1 3 2 1 0 0 0 1 0 1 0 3 2 4 1 0 0 0 0 0 0 3 4 1 0 1 0 0 0 1 0 3 4 2 1 0 0 0 0 0 0 4 1 2 0 0 1 0 0 0 0 4 1 3 0 1 0 0 0 0 0 4 1 6 0 0 0 0 1 0 1 4 2 1 0 0 1 0 0 1 0 4 2 3 1 0 0 0 0 0 0 4 3 1 0 1 0 0 0 1 0 4 3 2 1 0 0 0 0 0 0 5 6 1 0 1 1 1 0 0 0 6 1 2 0 0 1 1 0 0 0 6 1 3 0 1 0 1 0 0 0 6 1 4 0 1 1 0 0 0 0 7 6 1 0 1 1 1 0 0 0 ACPP(G2) 6 2 3 1 6 0 0 0 0 1 0 1 2 3 4 1 0 1 0 0 0 1 0 2 4 1 6 0 0 0 0 1 0 1 2 4 3 1 0 0 0 0 0 1 0 2 1 4 0 1 1 0 0 0 0 2 1 6 0 0 0 0 1 0 1 2 4 1 0 1 1 0 0 1 0 3 2 1 6 0 0 0 0 1 0 1 3 4 2 1 0 0 0 0 0 1 0 3 4 1 6 0 0 0 0 1 0 1 2 1 3 0 1 0 1 0 0 0 2 3 1 0 1 0 1 0 1 0 2 3 4 1 1 0 0 0 0 0 2 4 3 1 1 0 0 0 0 0 4 2 1 6 0 0 0 0 1 0 1 4 2 3 1 0 0 0 0 0 1 0 4 3 1 6 0 0 0 0 1 0 1 4 3 2 1 0 0 0 0 0 1 0 5 6 1 2 0 0 1 1 0 0 0 5 6 1 3 0 1 0 1 0 0 0 5 6 1 4 0 1 1 0 0 0 0 6 1 2 3 0 0 0 1 0 0 0 6 1 2 4 0 0 1 0 0 0 0 6 1 3 2 0 0 0 1 0 0 0 6 1 3 4 0 1 0 0 0 0 0 6 1 4 2 0 0 1 0 0 0 0 6 1 4 3 0 1 0 0 0 0 0 7 6 1 2 0 0 1 1 0 0 0 7 6 1 3 0 1 0 1 0 0 0 7 6 1 4 0 1 1 0 0 0 0 3 2 4 1 0 0 0 0 0 1 0 1 4 3 1 1 0 0 0 0 0 4 1 2 0 0 1 1 0 0 0 4 1 3 0 1 0 1 0 0 0 4 1 6 0 0 0 0 1 0 1 4 2 1 0 0 1 1 0 1 0 4 2 3 1 0 0 1 0 0 0 4 3 1 0 1 0 1 0 1 0 4 3 2 1 0 0 1 0 0 0 1 3 2 1 0 0 1 0 0 0 1 3 4 1 1 0 0 0 0 0 1 4 2 1 0 1 0 0 0 0 3 1 2 0 0 1 1 0 0 0 3 2 4 1 0 1 0 0 0 0 3 4 1 0 1 1 0 0 1 0 3 4 2 1 0 1 0 0 0 0 4 1 2 0 0 0 1 0 0 0 4 1 3 0 0 0 1 0 0 0 4 2 1 0 0 0 1 0 1 0 4 3 1 0 0 0 1 0 1 0 3 1 4 0 0 1 0 0 0 0 3 1 6 0 0 0 0 1 0 1 3 4 1 0 0 1 0 0 1 0 2 3 4 0 1 0 0 0 0 0 2 4 3 0 1 0 0 0 0 0 1 2 3 1 0 0 1 0 0 0 1 2 4 1 0 1 0 0 0 0 4 2 3 0 0 0 1 0 0 0 4 3 2 0 0 0 1 0 0 0 3 1 4 0 1 1 0 0 0 0 3 2 1 0 0 1 1 0 1 0 3 2 4 0 0 1 0 0 0 0 3 4 2 0 0 1 0 0 0 0 5 7 UCL(G2) Same sequence diff start/dir), same cycle 1 4 3 0 1 0 0 0 0 0 3 1 2 0 0 0 1 0 0 0 1 3 2 0 0 0 1 0 0 0 1 3 4 0 1 0 0 0 0 0 1 4 2 0 0 1 0 0 0 0 1 2 3 1 1 2 4 1 1 2 3 0 0 0 1 0 0 0 1 2 4 0 0 1 0 0 0 0 2 1 4 0 0 1 0 0 0 0 2 1 6 0 0 0 0 1 0 1 2 4 1 0 0 1 0 0 1 0 1 3 4 1 2 1 3 0 0 0 1 0 0 0 2 3 1 0 0 0 1 0 1 0 3 1 6 0 0 0 0 1 0 1 3 2 1 0 0 0 1 0 1 0 3 4 1 0 0 0 0 0 1 0 4 1 6 0 0 0 0 1 0 1 4 2 1 0 0 0 0 0 1 0 4 3 1 0 0 0 0 0 1 0 2 UACPP(G2) Remove path reverses 1 1 0 1 1 1 0 1 0 2 0 0 1 1 0 0 0 3 0 0 0 1 0 0 0 5 0 0 0 0 0 1 0 6 0 0 0 0 0 0 1 1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 2 3 1 6 0 0 0 0 1 0 1 2 3 4 1 0 1 0 0 0 1 0 2 4 1 6 0 0 0 0 1 0 1 2 4 3 1 0 0 0 0 0 1 0 3 2 1 6 0 0 0 0 1 0 1 3 4 1 6 0 0 0 0 1 0 1 3 4 2 1 0 0 0 0 0 1 0 4 2 1 6 0 0 0 0 1 0 1 4 2 3 1 0 0 0 0 0 1 0 4 3 1 6 0 0 0 0 1 0 1 4 3 2 1 0 0 0 0 0 1 0 3 2 4 1 0 0 0 0 0 1 0 1 2 0 0 1 1 0 0 0 1 3 0 1 0 1 0 0 0 1 4 0 1 1 0 0 0 0 1 6 0 0 0 0 1 0 1 2 1 0 0 1 1 0 1 0 2 3 0 0 0 1 0 0 0 2 4 0 0 1 0 0 0 0 3 1 0 0 0 1 0 1 0 3 2 0 0 0 1 0 0 0 4 1 0 0 0 0 0 1 0 5 6 0 0 0 0 0 0 1 4 3

  8. SG Clique Mining K=2: 2Cliques (2 vertices): 12 13 14 1623 24 34 56 67Find endptsof each edges (Int((n-1)/7)+1, Mod(n-1,7) +1) key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 2,1 2,2 2,3 2,4 2,5 2,6 2,7 3,1 3,2 3,3 3,4 3,5 3,6 3,7 4,1 4,2 4,3 4,4 4,5 4,6 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 6,1 6,2 6,3 6,4 6,5 6,6 6,7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 PE 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 2,1 2,2 2,3 2,4 2,5 2,6 2,7 3,1 3,2 3,3 3,4 3,5 3,6 3,7 4,1 4,2 4,3 4,4 4,5 4,6 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 6,1 6,2 6,3 6,4 6,5 6,6 6,7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 E 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 EU 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 C 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CU 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 3 4 5 6 7 8 9 k=3: 123 124 134 234 k=4: 1234 (123 124 234 are cliques) 123,1341234. 123.2341234. 124,1341234. 124, 2341234. 134,2341234. 1234 only 4-clique Using the EdgeCountthm: on C={1,2,3,4}, CU=C&EU C is a clique since ct(CU)=comb(4, 2)=4!/2!2!=6 k=2: E=12 13 14 16 23 2434 56 57 67. k=3: 123 124 134 234 567 8 EC, requires counting 1’s in mask pTree of each Subgraph (or candidate Clique, if take the time to generate the CCSs – but then clearly the fastest way to finish up is simply to lookup the single bit position in E, i.e., use EC). EdgeCount Algorithm (EC): |PUC| = (k+1)!/(k-1)!2! then CCCS The SG alg only needs Edge Mask pTree, E, and a fast way to find those pairs of subgraphs in CSk that share k-1 vertices (then check E to see if the two different kth vertices are an edge in G. Again this is a standard part of the Apriori ARM algorithm and has therefore been optimized and engineered ad infinitum!) PE(4,8)=1 2348CS4 PE(3,8)=1 1348CS4 PE(4,8)=1 1248CS4 PE(2,6)=0 PE(2,6)=0 key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8 8.1 8,2 8,3 8,4 8,5 8,6 8,7 8.8 E 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 6 G2 6 G4 6 G3 have PE(2,4)=1 1234CS4 PE(6,7)=1 567CS3 PE(6,7)=1 567CS3 PE(2,4)=1 124CS3 5 5 5 k=4: 1234 1238 1248 1348 2348 k=2: 12 13 14 16 23 2434 56 57 67 18 28 38 48. 7 7 7 PE(2,4)=1 124CS3 PE(1,5)=0 PE(1,5)=0 PE(2,3)=1 123CS3 PE(2,4)=1 1234CS4 PE(4,8)=1 148CS3 PE(2,3)=1 So 123CS3 already have 567 PE(1,7)=0 have PE(1,7)=0 PE(6,8)=0 have Have PE(3,8)=1 238CS3 k=5: 12348 = CS5. PE(2,8)=1 128CS3 PE(3,8)=1 138CS3 2 2 2 1 1 1 PE(2,3)=1 234CS3 PE(3,8)=1 1238CS4 PE(4,8)=1 248CS3 PE(4,8)=1 348CS3 PE(2,3)=1 234CS3 Have 123CS3 have have 124CS3 Have Have 1234 PE(1,4)=1 134CS3 PE(1,4)=1 134CS3 PE(4,8)=1 12348CS5 Have 4 3 4 4 3 3 k=3: 123 124 134 234 567 128 138 148 238 248 348

  9. TheEdgepTree(E), PathTree(PT), ShortestPathvTree(SPT),AcyclicPathTree(APT) andCycleList(CL) of a graph, G5 PT Clique Miner Algorithm A clique is all cycles Extend to a k-plex(k-core) mining alg? PT(=APT+CL), SPT are powerful datamining tools with closure properties (to eliminate branches) . PTG5 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 EG5 2-level str=8 PTG5 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 EG5 2-level str=8 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 3 6 0 0 0 0 0 0 0 1 1 2 0 0 0 1 0 0 0 0 1 5 0 0 0 0 0 0 1 0 1 7 0 0 0 0 1 0 0 0 2 1 0 0 0 0 1 0 1 0 3 8 0 0 0 0 0 1 0 0 4 2 1 0 0 0 0 0 0 0 5 1 0 1 0 0 0 0 1 0 5 7 1 0 0 0 0 0 0 0 6 3 0 0 0 0 0 0 0 1 6 8 0 0 1 0 0 0 0 0 7 1 0 1 0 0 1 0 0 0 7 5 1 0 0 0 0 0 0 0 8 3 0 0 0 0 0 1 0 0 8 6 0 0 1 0 0 0 0 0 k-plex (missingk edges) mine alg? k-core (has  k edges) mining alg? 3 6 0 0 0 0 0 0 0 1 1 2 0 0 0 1 0 0 0 0 1 5 0 0 0 0 0 0 1 0 1 7 0 0 0 0 1 0 0 0 2 1 0 0 0 0 1 0 1 0 3 8 0 0 0 0 0 1 0 0 4 2 1 0 0 0 0 0 0 0 5 1 0 1 0 0 0 0 1 0 5 7 1 0 0 0 0 0 0 0 6 3 0 0 0 0 0 0 0 1 6 8 0 0 1 0 0 0 0 0 7 1 0 1 0 0 1 0 0 0 7 5 1 0 0 0 0 0 0 0 8 3 0 0 0 0 0 1 0 0 8 6 0 0 1 0 0 0 0 0 k-plex (missingk edges) mine alg? k-core (has  k edges) mining alg? 1 5 7 1 0 0 0 0 0 0 0 1 7 5 1 0 0 0 0 0 0 0 2 1 5 0 0 0 0 0 0 1 0 2 1 7 0 0 0 0 1 0 0 0 3 6 8 0 0 1 0 0 0 0 0 3 8 6 0 0 1 0 0 0 0 0 4 2 1 0 0 0 0 1 0 1 0 5 1 2 0 0 0 1 0 0 0 0 5 1 7 0 0 0 0 1 0 0 0 5 7 1 0 0 0 0 1 0 0 0 6 3 8 0 0 0 0 0 1 0 0 6 8 3 0 0 0 0 0 1 0 0 7 1 2 0 0 0 1 0 0 0 0 7 1 5 0 0 0 0 0 0 1 0 8 6 3 0 0 0 0 0 0 0 1 8 3 6 0 0 0 0 0 0 0 1 7 5 1 0 1 0 0 0 0 1 0 Density (internal edge density >>external|avg) mining alg? Degree (internal vertex degree >> external|avg) mining alg? 1 5 7 1 0 0 0 0 0 0 0 1 7 5 1 0 0 0 0 0 0 0 2 1 5 0 0 0 0 0 0 1 0 2 1 7 0 0 0 0 1 0 0 0 3 6 8 0 0 1 0 0 0 0 0 3 8 6 0 0 1 0 0 0 0 0 4 2 1 0 0 0 0 1 0 1 0 5 1 2 0 0 0 1 0 0 0 0 5 1 7 0 0 0 0 1 0 0 0 5 7 1 0 0 0 0 1 0 0 0 6 3 8 0 0 0 0 0 1 0 0 6 8 3 0 0 0 0 0 1 0 0 7 1 2 0 0 0 1 0 0 0 0 7 1 5 0 0 0 0 0 0 1 0 8 6 3 0 0 0 0 0 0 0 1 8 3 6 0 0 0 0 0 0 0 1 7 5 1 0 1 0 0 0 0 1 0 Density (internal edge density >>external|avg) mining alg? Degree (internal vertex degree >> external|avg) mining alg? 4 2 1 5 0 0 0 0 0 0 1 0 4 2 1 7 0 0 0 0 1 0 0 0 7 5 1 2 0 0 0 1 0 0 0 0 Max clique MiningA kCycle is a kCliqueiff it’s found in CLk as PERM(k-1,k-1)/2=(k-1)!/2 kCycles (e.g., vertices repeated in CL for 3cycles, 2!/2=1; 4cycles, 3!/2=3; 5cycles, 4!/2=12; 6cycles, 5!/2=60. 4 2 1 5 0 0 0 0 0 0 1 0 4 2 1 7 0 0 0 0 1 0 0 0 7 5 1 2 0 0 0 1 0 0 0 0 1 1 2 2 Downward closure: Once, a 4cycle 12341 is established as a 4clique (by the fact that {1,2,3,4} occurs 3!/2=3 times in CL), all 3vertex subsets are 3cliques {1,2,3},{1,2,4},{1,3,4}, so no need to check further. 3 3 8 8 CLG5 1571 APTG5 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 1751 4 4 7 7 3683 DiamG5 is max{Diamk} = max{ 22132131}=3. Connect comp containing V1, COMP1={1,2,4,5,7}. 1st vertexCOMP1,3, COMP3 ={3,6,8}. Done. Partition={ {1,2,4,5,7}, {3,6,8} }. To pick the first vertexCOMP1, mask off COMP1 with SPTv1’, pick 1st vertex in this complement. CLG5 1571 APTG5 5 5 6 6 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 3863 1751 5175 3683 5715 2 1 0 0 0 0 1 0 1 0 3 6 0 0 0 0 0 0 0 1 3 8 0 0 0 0 0 1 0 0 4 2 1 0 0 0 0 0 0 0 5 1 0 1 0 0 0 0 1 0 5 7 1 0 0 0 0 0 0 0 6 3 0 0 0 0 0 0 0 1 6 8 0 0 1 0 0 0 0 0 7 1 0 1 0 0 1 0 0 0 7 5 1 0 0 0 0 0 0 0 8 3 0 0 0 0 0 1 0 0 8 6 0 0 1 0 0 0 0 0 1 2 0 0 0 1 0 0 0 0 1 5 0 0 0 0 0 0 1 0 1 7 0 0 0 0 1 0 0 0 3863 6386 5175 6836 5715 2 1 0 0 0 0 1 0 1 0 3 6 0 0 0 0 0 0 0 1 3 8 0 0 0 0 0 1 0 0 4 2 1 0 0 0 0 0 0 0 5 1 0 1 0 0 0 0 1 0 5 7 1 0 0 0 0 0 0 0 6 3 0 0 0 0 0 0 0 1 6 8 0 0 1 0 0 0 0 0 7 1 0 1 0 0 1 0 0 0 7 5 1 0 0 0 0 0 0 0 8 3 0 0 0 0 0 1 0 0 8 6 0 0 1 0 0 0 0 0 1 2 0 0 0 1 0 0 0 0 1 5 0 0 0 0 0 0 1 0 1 7 0 0 0 0 1 0 0 0 7157 6386 7517 6836 8368 2 1 5 0 0 0 0 0 0 1 0 2 1 7 0 0 0 0 1 0 0 0 4 2 1 0 0 0 0 1 0 1 0 5 1 2 0 0 0 1 0 0 0 0 7 1 2 0 0 0 1 0 0 0 0 7 5 1 0 1 0 0 0 0 0 0 7157 8638 7517 8368 2 1 5 0 0 0 0 0 0 1 0 2 1 7 0 0 0 0 1 0 0 0 4 2 1 0 0 0 0 1 0 1 0 5 1 2 0 0 0 1 0 0 0 0 7 1 2 0 0 0 1 0 0 0 0 7 5 1 0 1 0 0 0 0 0 0 SPTG5 8638 1 0 1 0 0 1 0 1 0 2 1 0 0 1 2 0 2 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 4 2 1 0 0 3 0 3 0 5 1 2 0 3 0 0 1 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 7 1 2 0 0 1 0 0 0 7 1 2 0 3 1 0 0 0 8 0 0 1 0 0 1 0 0 1 0 1 0 2 1 0 1 0 5 1 2 0 0 0 0 1 0 4 2 1 0 0 0 0 0 0 SPTG5 4 2 1 5 0 0 0 0 0 0 1 0 4 2 1 7 0 0 0 0 1 0 0 0 7 5 1 2 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 2 1 0 0 1 2 0 2 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 4 2 1 0 0 3 0 3 0 5 1 2 0 3 0 0 1 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 7 1 2 0 0 1 0 0 0 7 1 2 0 3 1 0 0 0 8 0 0 1 0 0 1 0 0 1 0 1 0 2 1 0 1 0 5 1 2 0 0 0 0 1 0 4 2 1 0 0 0 0 0 0 4 2 1 5 0 0 0 0 0 0 1 0 4 2 1 7 0 0 0 0 1 0 0 0 7 5 1 2 0 0 0 1 0 0 0 0 DiamG5 is max{Diamk} = max{ 2,2,1,3,2,1,3,1}=3. Connected comp containing V1, COMP1={1,2,4,5,7}. Pick 1st vertex not in COMP1,3, COMP3 ={3,6,8}. Done. The partition is { {1,2,4,5,7}, {3,6,8} }. To pick the first vertex not in COMP1, mask off COMP1 with SPTv1’ and then pick the first vertex in this complement.

  10. 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 SP1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 SP1&2 The EdgepTree(E), PathTree(PT), ShortestPathvTree(SPT),AcyclicPathTree(APT) andCycleList(CL) of G5 a e b 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 9 d 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 b f g c A6Ps A5Ps A4Ps A2Ps 8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 E=A1Ps 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 A3Ps 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 SP1&2&3 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 SP2 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 6 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 cycles in blue (not in APT) A c 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 D g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 G f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 b c 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 D f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 F d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 F g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 G d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 4 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 6 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 6 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 8 4 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 9 c 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 SP1&2&3&4 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 SP3 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 8 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 8 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 4 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 4 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 4 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 4 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 5 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 5 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 5 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 6 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 6 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 6 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 7 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 7 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 7 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 7 1 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 5 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 5 7 6 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 5 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 8 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G F d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 D G f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 6 7 5 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 5 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 D F g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 F D g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 F G d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6 5 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 G D f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 4 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 7 6 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 SP1&2&3&4&5 COMPLETE 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 SP4 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 1 2 4 3 1 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 4 3 1 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 5 7 6 1 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 5 6 1 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 4 3 1 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 4 3 1 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 4 3 1 6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 4 3 1 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 3 1 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 5 6 1 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 7 6 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 7 6 1 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 5 6 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 4 3 1 6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 2 4 3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 1 6 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 1 6 7 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 3 1 6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 6 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 7 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 3 4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 5 6 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 6 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 4 3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 SP5 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 SP6 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 G6

  11. SPT: Shortest Path Tree EdgepTree(E), PathTree(PT) ShortestPathvTree(SPT),AcyclicPathTree(APT) andCycleList(CL) of G6 G6 PT1 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 4 1 1 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 6 0 0 0 0 1 0 1 1 0 0 0 0 7 0 0 0 1 1 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 9 0 0 0 0 0 0 0 1 0 1 1 1 c 0 0 1 0 0 0 0 0 1 1 1 0 8 0 0 0 0 0 1 0 0 1 1 0 0 a 0 0 0 0 0 0 0 1 1 0 1 1 8 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 4 0 0 1 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 1 0 0 0 0 0 4 0 0 1 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 1 1 0 0 0 0 0 5 1 1 0 0 0 0 0 5 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 1 1 0 0 0 0 0 6 1 1 0 0 0 0 6 1 1 0 0 0 0 6 0 0 0 0 0 0 6 1 1 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 SP4 1 2 3 4 5 6 7 8 9 a b c SP 1 2 3 4 5 6 7 8 9 a b c SP3 1 2 3 4 5 6 7 8 9 a b c SP1 1 2 3 4 5 6 7 8 9 a b c 9 1 1 1 9 1 1 1 9 0 0 0 9 0 0 0 9 0 0 0 9 1 1 1 8 0 0 0 0 8 1 1 0 0 8 1 1 0 0 8 0 0 0 0 8 1 1 0 0 a 1 1 a 1 1 a 0 0 a 1 1 a 0 0 a 0 0 b 0 b 1 b 0 b 1 b 1 b 0 1 2 3 4 5 6 7 8 9 a b c 2 0 0 0 0 1 0 0 0 0 1 3 1 0 0 0 0 1 1 1 0 4 1 1 0 0 0 0 0 0 5 0 0 1 0 0 0 0 7 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 6 0 0 1 1 0 0 8 0 0 1 1 C 3 1 1 0 0 0 0 0 0 0 0 0 0 A b 0 0 0 0 0 0 0 0 1 0 0 1 B 9 0 0 0 0 0 0 0 1 0 1 0 1 B c 0 0 1 0 0 0 0 0 1 1 0 0 A 9 0 0 0 0 0 0 0 1 0 0 1 1 A c 0 0 1 0 0 0 0 0 1 0 1 0 B a 0 0 0 0 0 0 0 1 1 0 0 1 C 9 0 0 0 0 0 0 0 1 0 1 1 0 3 1 0 1 0 1 0 0 0 0 0 0 0 0 3 2 1 0 0 1 0 0 0 0 0 0 0 0 3 c 0 0 0 0 0 0 0 0 1 1 1 0 6 5 0 0 0 0 0 0 1 0 0 0 0 0 6 7 0 0 0 1 1 0 0 0 0 0 0 0 8 6 0 0 0 0 1 0 1 0 0 0 0 0 A 8 0 0 0 0 0 1 0 0 1 0 0 0 C a 0 0 0 0 0 0 0 1 1 0 1 0 9 b 0 0 0 0 0 0 0 0 0 1 0 1 4 7 0 0 0 0 1 1 0 0 0 0 0 0 5 6 0 0 0 0 0 0 1 1 0 0 0 0 C b 0 0 0 0 0 0 0 0 1 1 0 0 1 2 0 0 1 1 0 0 0 0 0 0 0 0 1 3 0 1 0 0 0 0 0 0 0 0 0 1 1 4 0 1 0 0 0 0 1 0 0 0 0 0 2 3 1 0 0 0 0 0 0 0 0 0 0 1 2 4 1 0 0 0 0 0 1 0 0 0 0 0 8 9 0 0 0 0 0 0 0 0 0 1 1 1 9 c 0 0 1 0 0 0 0 0 0 1 1 0 4 1 0 1 1 0 0 0 0 0 0 0 0 0 4 2 1 0 1 0 0 0 0 0 0 0 0 0 5 7 0 0 0 1 0 1 0 0 0 0 0 0 6 8 0 0 0 0 0 0 0 0 1 1 0 0 7 4 1 1 0 0 0 0 0 0 0 0 0 0 7 5 0 0 0 0 0 1 0 0 0 0 0 0 7 6 0 0 0 0 1 0 0 1 0 0 0 0 8 a 0 0 0 0 0 0 0 0 1 0 1 1 9 8 0 0 0 0 0 1 0 0 0 1 0 0 9 a 0 0 0 0 0 0 0 1 0 0 1 1 PT2 1 2 3 4 5 6 7 8 9 a b c SPT2 SPT3 SPT4 2 3 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 1 0 0 0 0 1 2 4 0 0 0 0 1 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 3 c 0 0 0 0 0 1 1 1 0 4 7 1 1 0 0 0 0 0 0 5 6 0 0 1 0 0 0 0 6 8 0 0 1 1 0 0 7 6 1 0 0 0 0 8 9 0 0 1 1 1 2 3 4 5 6 7 8 9 a b c 1 2 3 4 5 6 7 8 9 a b c 1 2 3 4 5 6 7 8 9 a b c 3 C a 0 0 0 0 0 0 0 1 1 0 1 0 1 2 3 1 0 0 0 0 0 0 0 0 0 0 1 1 2 4 1 0 0 0 0 0 1 0 0 0 0 0 1 3 c 0 0 0 0 0 0 0 0 1 1 1 0 1 4 2 1 0 1 0 0 0 0 0 0 0 0 0 1 4 7 0 0 0 0 1 1 0 0 0 0 0 0 2 4 1 0 1 1 0 0 0 0 0 0 0 0 0 2 4 7 0 0 0 0 1 1 0 0 0 0 0 0 3 1 2 0 0 1 1 0 0 0 0 0 0 0 0 3 1 4 0 1 0 0 0 0 1 0 0 0 0 0 4 1 2 0 0 1 1 0 0 0 0 0 0 0 0 4 1 3 0 1 0 0 0 0 0 0 0 0 0 1 4 2 1 0 0 1 1 0 0 0 0 0 0 0 0 4 2 3 1 0 0 0 0 0 0 0 0 0 0 1 4 7 5 0 0 0 0 0 1 0 0 0 0 0 0 4 7 6 0 0 0 0 1 0 0 1 0 0 0 0 5 6 8 0 0 0 0 0 0 0 0 1 1 0 0 5 7 4 1 1 0 0 0 0 0 0 0 0 0 0 5 7 6 0 0 0 0 1 0 0 1 0 0 0 0 6 5 7 0 0 0 1 0 1 0 0 0 0 0 0 6 7 4 1 1 0 0 0 0 0 0 0 0 0 0 6 7 5 0 0 0 0 0 1 0 0 0 0 0 0 7 4 1 0 1 1 0 0 0 0 0 0 0 0 0 7 4 2 1 0 1 0 0 0 0 0 0 0 0 0 3 C b 0 0 0 0 0 0 0 0 1 1 0 0 5 6 7 0 0 0 1 1 0 0 0 0 0 0 0 6 8 9 0 0 0 0 0 0 0 0 0 1 1 1 7 5 6 0 0 0 0 0 0 1 1 0 0 0 0 7 6 5 0 0 0 0 0 0 1 0 0 0 0 0 1 3 2 1 0 0 1 0 0 0 0 0 0 0 0 2 3 1 0 1 0 1 0 0 0 0 0 0 0 0 6 8 a 0 0 0 0 0 0 0 0 1 0 1 1 3 C 9 0 0 0 0 0 0 0 1 0 1 1 0 2 3 c 0 0 0 0 0 0 0 0 1 1 1 0 7 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 0 0 1 1 0 0 1 1 1 0 3 0 0 0 1 1 0 0 0 0 3 0 0 0 1 0 0 0 0 0 4 0 0 0 1 0 0 0 1 5 0 0 0 1 1 0 0 6 0 0 0 0 1 1 7 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 PT3 1 2 3 4 5 6 7 8 9 a b c … 1 3 c 0 0 0 0 0 0 0 1 1 1 0 1 4 7 0 0 0 1 1 0 0 0 0 0 0 2 3 c 0 0 0 0 0 0 0 1 1 1 0 2 4 7 0 0 0 1 1 0 0 0 0 0 0 3 1 4 0 0 0 1 0 0 0 0 0 3 C 9 0 0 0 0 1 0 0 0 0 4 1 3 0 0 0 0 0 0 0 1 5 6 8 0 0 0 1 1 0 0 4 7 6 0 0 0 1 0 0 0 0 6 8 9 0 0 0 0 1 1 7 6 8 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 3 0 1 1 0 0 0 0 0 0 4 0 0 0 0 1 1 1 0 5 0 0 0 0 0 1 1 7 0 0 0 1 1 5 0 0 0 0 0 0 0 7 0 0 0 0 0 1 3 C 9 0 0 0 0 0 0 0 1 0 1 1 0 5 6 8 9 0 0 0 0 0 0 0 0 0 1 1 1 2 3 C 9 0 0 0 0 0 0 0 1 0 1 1 0 3 1 4 7 0 0 0 0 1 1 0 0 0 0 0 0 4 1 3 c 0 0 0 0 0 0 0 0 1 1 1 0 7 6 8 9 0 0 0 0 0 0 0 0 0 1 1 1 PT4 1 2 3 4 5 6 7 8 9 a b c 1 5 … … … … … … … 4 2 6 7 3 c 9 b 8 a 1 3 C 9 0 0 0 0 0 0 1 0 0 0 0 3 1 4 7 0 1 1 0 0 0 0 0 0 2 3 C 9 0 0 0 0 0 0 1 0 0 0 0 4 1 3 c 0 0 0 0 1 1 1 0 5 6 8 9 0 0 0 0 0 1 1 7 6 8 9 0 0 0 0 0 1 1 1 1 1 1 3 3 2 4 3 3 3 2 1 1 1 1 3 3 2 0 3 3 3 2 2 1 1 3 3 2 4 3 3 3 2 2 1 1 3 3 2 0 3 3 3 2 3 2 0 0 0 0 2 2 2 1 1 1 1 1 0 0 2 0 0 0 0 2 2 1 1 0 0 2 0 0 0 0 2 3 2 0 0 3 3 2 2 2 1 3 2 0 0 3 0 2 2 2 1 4 2 2 1 0 0 0 0 0 5 1 1 2 0 0 0 0 6 1 1 2 2 0 0 7 2 0 0 0 0 3 2 4 4 3 3 2 2 2 1 5 1 1 2 3 3 4 4 4 2 2 1 3 0 0 0 3 5 1 1 2 3 3 0 0 6 1 1 2 2 3 3 7 2 3 3 0 0 7 2 3 3 4 4 8 1 1 2 2 4 2 2 1 3 4 4 4 3 CycleList 1231 1241 5675

  12. 1 2 1 3 1 4 2 1 2 3 2 4 3 1 3 2 3 C 4 1 4 2 4 7 5 6 5 7 6 5 6 7 6 8 7 4 7 5 7 6 B 9 B A B C 9 8 9 A 9 B 9 C C 3 C 9 C A C B 8 6 8 9 8 A A 8 A 9 A B A C E1=SP1=PT1 1 0 1 1 1 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 0 0 0 0 0 0 3 3 1 1 0 0 0 0 0 0 0 0 0 1 3 4 1 1 0 0 0 0 1 0 0 0 0 0 3 5 0 0 0 0 0 1 1 0 0 0 0 0 2 6 0 0 0 0 1 0 1 1 0 0 0 0 3 7 0 0 0 1 1 1 0 0 0 0 0 0 3 b 0 0 0 0 0 0 0 0 1 1 0 1 3 9 0 0 0 0 0 0 0 1 0 1 1 1 4 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 0 0 0 1 0 0 1 1 0 0 3 a 0 0 0 0 0 0 0 1 1 0 1 1 4 SP1 1 2 3 4 5 6 7 8 9 a b c 38 1 5 2 4 6 7 G6 3 1 2 3 1 2 4 1 3 2 1 3 C 1 4 8 c b 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 b 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 SP2 1 2 3 4 5 6 7 8 9 a b c 9 a 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 b 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 SP3 1 2 3 4 5 6 7 8 9 a b c

  13. 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16=1deg 9 13 19 16 13 12 13 17 24 19 14 25 14 25 15 15 3 15 16 26 15 16 16 15 6 6 13 20 21 15 20 26 11 6=2dg 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP1 10,25,26,28,29,33,34 not shown (only 17 on, 1=4dg) 15,16,19,21,23,24,27,30 only 17 on, 5deg=1 G7 17 SP5 5 6 7 11 2 3 5 6 7 8 9 21 2 3 4 7 30 SP4 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 11 4 11 8 8 8 12 3 11 8 8 9 3 6 6 12 8 6 4 6 8 6 4 23 23 6 7 8 5 8 1 10 10=3dg 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8=4dg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 8=5dg 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 3 1234567890123456789012345678901234 ver g9a63444523125222223222533243446bg 1dg 9djgdcdhojepepff3fgqfggf66dklfkqb6 2dg 8b4b888c3b889366c8646864nn678581aa 3dg 000088800188809a8880888811a1180011 4dg 0000000000000011801010110010010000 5dg 17 is an outlier. Try clustering by SPdeg from 17. The SPk17 pTrees mask the clustering (next slide) EdgepTree(E), PathTree(PT) ShortestPathvTree(SPT),AcyclicPathTree(APT) andCycleList(CL) of G7 BASE 65 1 2 3 4 5 6 01234567890123456789012345678901234567890123456789012345678901234 0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ@#$

  14. Shortest Path Trees Construction (We don’t need the Path Trees to get the Shortest Path Trees! That’s because a subpath of a shortest path is a shortest path.) SPSF12 SPSF13 SPSF11 SPSF1’2 SPSF1’3 SPSF1’1 G6 S1P=E 2 1 1 1 1 0 0 0 0 0 0 0 0 2 0 0 0 0 1 1 1 1 1 1 1 1 3 1 1 1 0 0 0 0 0 0 0 0 1 3 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 4 1 1 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 6 0 0 0 0 1 0 1 1 0 0 0 0 7 0 0 0 1 1 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 9 0 0 0 0 0 0 0 1 0 1 1 1 c 0 0 1 0 0 0 0 0 1 1 1 0 8 0 0 0 0 0 1 0 0 1 1 0 0 a 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 SPSF23 SPSF21 SPSF2’3 SPSF2’1 S2P2=SPSF1’2&(ORjS1P2Ej ) S2P3=SPSF1’3&(ORjS1P3Ej ) S2P1=SPSF1’1&(ORjS1P1Ej ) 2 1 0 1 1 0 0 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 4 1 1 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 4 1 1 0 0 0 0 1 0 0 0 0 0 3 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 0 0 0 0 0 c 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 S2P 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 0 1 3 1 1 1 1 0 0 0 0 1 1 1 1 4 0 0 1 0 1 1 0 0 0 0 0 0 5 0 0 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 1 3 0 0 0 1 0 0 0 0 1 1 1 0 6 0 0 0 1 0 0 0 0 1 1 0 0 3 0 0 0 0 1 1 1 1 0 0 0 0 7 1 1 0 0 0 0 0 1 0 0 0 0 8 0 0 0 0 1 0 1 0 0 0 1 1 9 0 0 1 0 0 1 0 0 0 0 0 0 a 0 0 1 0 0 1 0 0 0 0 0 0 b 0 0 1 0 0 0 0 1 0 0 0 0 c 1 1 0 0 0 0 0 1 0 0 0 0 Identical to 1 from here on. SPSF33 SPSF31 SPSF3’1 SPSF3’3 S3P3=SPSF2’3&(ORjS2P3Ej ) S3P1=SPSF2’1&(ORjS2P1Ej ) 7 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 c 0 0 1 0 0 0 0 0 1 1 1 0 S3P 3 0 0 0 0 1 1 1 1 0 0 0 0 4 1 1 0 0 0 0 1 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 9 0 0 0 0 0 0 0 1 0 1 1 1 a 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 3 1 1 1 1 0 0 1 1 1 1 1 1 3 0 0 0 0 1 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 1 5 1 1 0 0 0 0 0 0 1 1 0 0 2 0 0 0 0 1 1 0 0 1 1 1 0 3 0 0 0 0 0 0 1 1 0 0 0 0 6 1 1 0 0 0 0 0 0 0 0 1 1 7 0 0 1 0 0 0 0 0 1 1 0 0 8 0 0 1 1 0 0 0 0 0 0 0 0 9 1 1 0 0 1 0 1 0 0 0 0 0 a 1 1 0 0 1 0 1 0 0 0 0 0 b 1 1 0 0 0 1 0 0 0 0 0 0 c 0 0 0 1 0 1 0 0 0 0 0 0 SPSF41 SPSF43 SPSF4’3 SPSF4’1 1 S4P3=SPSF3’3&(ORjS3P3Ej ) S4P1=SPSF3’1&(ORjS3P1Ej ) 5 What is the cost of creating the SPs? vV, there are ~Avg{Diam(v)vV} steps, each costs 1 complement of SPSF (cost =compl), OR of ~Avg|Ek| pTrees (cost=OrAvg|Ek| 1 SPSF & above_OR_result (cost=AND), 1 OR to update SPSF (cost=OR) Cost= |V|*AvgDiam*(compl+OR*AD+AND+OR), so O(|V|). I.e., linear in # of vertices, assuming AD=AvgDeg is small. This is a one-time, parallelizable construction over the vertices. For Friends, it is B*4*(3*pTOP+AD*pTOP)=4B*(3+AD)pTOP=B*pTOP*(12+4AD), where pTOP is the cost of a pTree Operation (comp, &, OR) and B=billion). Parallelized over an n node cluster, this 1-time Shortest Path Tree construction cost would be B*pTOP*(12+4AvgDeg) / n. The SnP’s capture only the shortest path lengths between all pairs of vertices. We could (have) capture actual shortest paths (all shortest paths?, all paths in PTs?), since we construct (but do not retain) that info along the way. How to structure it/index it?/residualize it? S4P 4 2 6 7 1 0 0 0 0 0 0 0 1 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 6 0 0 0 0 1 0 1 1 0 0 0 0 9 0 0 0 0 0 0 0 1 0 1 1 1 a 0 0 0 0 0 0 0 1 1 0 1 1 b 0 0 0 0 0 0 0 0 1 1 0 1 3 0 0 0 0 1 1 0 0 0 0 0 0 7 0 0 0 1 1 1 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 1 1 0 0 3 c 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 3 0 0 0 0 1 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 1 1 1 0 5 0 0 1 0 0 0 0 0 0 0 1 1 6 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 1 1 8 1 1 0 0 0 0 0 0 0 0 0 0 9 0 0 0 1 0 0 0 0 0 0 0 0 a 0 0 0 1 0 0 0 0 0 0 0 0 b 0 0 0 1 1 0 1 0 0 0 0 0 c 0 0 0 0 1 0 1 0 0 0 0 0 Done with Vertex 1 Shortest Paths. Diam(1)=4 Done with Vertex 3 Shortest Paths. Vertices 4-c SPs done the same way 9 b 8 a SPSF1i= S1Pi OR Mi , Mi has 1 only at i SPSF(k+1)i= SPSFki OR S(k+1)Pi S(k+1)Pi=SPSFk’i&(ORjSkPj Ej ) “The mask pTree of the shortest k+1 path starting at vertex i is the Shortest Paths So Far Complement ANDed with the OR of ith edge pTrees over all ithe Shortest k Path List”

  15. 17 is an outlier. Try clustering by SPdeg from 17. The SPk17 pTrees mask the clustering. SPdegk(17) 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 3 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 SPdeg=5: 15 16 19 21 23 24 27 30 SPdeg=4: 10 25 26 28 29 31 33 34 SPdeg=3: 2 3 4 8 9 12 13 14 18 20 22 32 SPdeg=2: 1 5 11 SPdeg=1: 6 7 G7 Now we would want to make this divisive and recursive. The maroon cluster could be broken apart into white and blue. Then one could use DegreeDifference within clusters to trade vertices among clustes to improve the DegDif quality measure. Maybe an agglomerative or divisive approach using SPdeg? Agglomerate two pieces together iff the SPdegdif is improved (or still exceeds a threshold?)? One could use Genetic Algorithm Hill Climbing to optimize clustering based on GAs applied to the SPdeg arrays. The bottom line is that there is a wealth of value in ShortestPathDegrees. One can easily mask subsets and recalculate SPdeg.

  16. 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16=1deg 9 13 19 16 13 12 13 17 24 19 14 25 14 25 15 15 3 15 16 26 15 16 16 15 6 6 13 20 21 15 20 26 11 6=2dg 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP1 10,25,26,28,29,33,34 not shown (only 17 on, 1=4dg) 15,16,19,21,23,24,27,30 only 17 on, 5deg=1 G7 17 SP5 5 6 7 11 2 3 5 6 7 8 9 21 2 3 4 7 30 SP4 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 11 4 11 8 8 8 12 3 11 8 8 9 3 6 6 12 8 6 4 6 8 6 4 23 23 6 7 8 5 8 1 10 10=3dg 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8=4dg 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 8=5dg 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 SP3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 and 34 have highest SP1deg (most siblings) at 16. Start with clusters, S(1), S(34) of siblings. Break ties with DegreeDiffs defined below. intdegS(x)=#edges from x to S-vertices. extdegS(x)=#edges from x to S’-vertices. DegDifS(x)=indegS(x)-extdegS(x) (or intdegS(x)/1+extdegS(x)? Start with S (and T,U,… if there are ties) =siblings of x of highest SP1degree. So for G7, S=Sibl(1) and T=Sibl(34). Add y(S’-T) to S iff DegDifS(y)>thresh1 and subract zS from S iff DegDif(z)<thesh2.

  17. G6 K-plex Search on G6: A k-plex is a Subgraph missing  k edges. All subgraphs will be induced subgraphs defined by their vertex set. Subgraph S has |ES|=s edges, |VS|=v vertices. S is a kplex iff C(v,2) – s = v(v-1)/2-s  k If S is a kplex, S’ adds 1 vertex, x to S, (V(S’)=V(S)!{x}) then S’ a kplex iff (v+1)v/2 – (deg(x,S’)+s)  k. Edges are 1-plexes. |E{123}| = |PE123| = 3 so 123 is a 0plex(clique) and a 1plex |E{124}| = |PE124| = 3 so 124 is a 0plex (clique) SP1=E 1 2 3 4 5 6 7 8 9 a b c SP3 1 2 3 4 5 6 7 8 9 a b c SP4 1 2 3 4 5 6 7 8 9 a b c SP2 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 1 0 2 1 0 1 1 0 0 0 0 0 0 0 0 3 2 0 0 0 0 0 0 1 0 0 0 0 1 2 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 1 1 0 0 1 1 1 0 3 1 1 0 0 0 0 0 0 0 0 0 1 3 3 0 0 0 1 0 0 0 0 1 1 1 0 3 0 0 0 0 0 0 1 1 0 0 0 0 3 0 0 0 0 1 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 1 4 0 0 0 0 0 0 0 0 1 1 1 0 4 1 1 0 0 0 0 1 0 0 0 0 0 3 4 0 0 1 0 1 1 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 0 0 1 1 5 1 1 0 0 0 0 0 0 1 1 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 2 5 0 0 0 1 0 0 0 1 0 0 0 0 6 0 0 0 0 1 0 1 1 0 0 0 0 3 6 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 1 0 0 0 0 1 1 0 0 6 1 1 0 0 0 0 0 0 0 0 1 1 7 1 1 0 0 0 0 0 1 0 0 0 0 7 0 0 0 1 1 1 0 0 0 0 0 0 3 7 0 0 1 0 0 0 0 0 1 1 0 0 7 0 0 0 0 0 0 0 0 0 0 1 1 b 0 0 0 1 1 0 1 0 0 0 0 0 b 1 1 0 0 0 1 0 0 0 0 0 0 b 0 0 1 0 0 0 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 3 9 1 1 0 0 1 0 1 0 0 0 0 0 9 0 0 1 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 1 1 1 4 9 0 0 0 1 0 0 0 0 0 0 0 0 c 0 0 0 0 1 0 1 0 0 0 0 0 c 1 1 0 0 0 0 0 1 0 0 0 0 c 0 0 0 1 0 1 0 0 0 0 0 0 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 0 0 0 1 0 0 1 1 0 0 3 8 0 0 0 0 1 0 1 0 0 0 1 1 8 1 1 0 0 0 0 0 0 0 0 0 0 8 0 0 1 1 0 0 0 0 0 0 0 0 a 0 0 0 1 0 0 0 0 0 0 0 0 a 0 0 1 0 0 1 0 0 0 0 0 0 a 1 1 0 0 1 0 1 0 0 0 0 0 a 0 0 0 0 0 0 0 1 1 0 1 1 4 If H is an ISG, |VH|=h, |EH|=H, H=h(h-1)/2 then H is a kplex iff H – H  k.. If H is a kplex and F is an ISG of H, then F is a kplex (if F is missing an edge than H is missing that edge also, since K inherits all H edges involving its vertices. F cannot be missing more edges than H.) If G isn’t a kplex, F1 an ISG of G with a vertex of least degree removed. If F1 isn’t a kplex, F2 ISG with a vertex of least degree removed, etc. until we find Fj to be a kplex. Remove Fj Repeat until all vertexes removed. We did a k-plex search of G6 by simple calculating edge counts (which are simply 1-counts of ANDed pTrees) using only SP1=E. G=12*11/2=66. G=19 G is a kplex for k  47. H1=ISG{12346789abc} (deg5=2). H1=11*10/2=55, H1=17. H1 is a kplex for k  37. H2=ISG{1234789abc} (deg6=2). H2=10*9/2=45, H2=15. H2 is a kplex for k  30. H3=ISG{123489abc} (deg7=1). H3=9*8/2=36, H3=14. H3 is a kplex for k  22. H4=ISG{12389abc} (deg4=2). H4=8*7/2=28, H4=12. H4 is a kplex for k  16. H5=ISG{1239abc} (deg8=2). H5=7*6/2=21, H5=10. H5 is a kplex for k  11. H6=ISG{239abc} (deg1=2). H6=6*5/2=15, H6=8. H6 is a kplex for k  7. H7=ISG{39abc} (deg2=1). H7=5*4/2=10, H7=7. H7 is a kplex for k  3. H8=ISG{9abc} (deg3=1). H8=4*3/2=6, H8=6. H8 is a kplex for k  0. So take out {9abc} and start over. G={12345678} G=8*7/2=28. G=10 G is a kplex for k  18. deg=33322331 H1=ISG{1234567} (deg8=1). H1=7*6/2=21, H1=9. H1 is a kplex for k  12. deg=2223223 1 5 H2=ISG{234567} (deg1=2). H2=6*5/2=15, H2=6. H2 is a kplex for k  9. deg=112223 4 2 6 7 3 H3=ISG{34567} (deg2=1). H3=5*4/2=10, H3=4. H3 is a kplex for k  6. deg=01222 c 9 b H4=ISG{4567} (deg3=0). H4=4*3/2=6, H4=4. H4 is a kplex for k  2. deg=1222 8 a So take out {567} and start over. H5=ISG{567} (deg4=1). H5=3*2/2=3, H5=3. H5 is a kplex for k  0. deg=222 G={12348} G=5*4/2=10. G=5 G is a kplex for k  5. deg=33220 H1=ISG{1234} (deg8=0). H1=4*3/2=6, H1=5. H1 is a kplex for k  1. deg=3322 H2=ISG{124} (deg3=2). H2=3*2/2=3, H2=3. H2 is a kplex for k  0. deg=222 This is exactly what we want ! 1234 is a 1plex (missing only 1 edge) and 124 was determined to be a clique (0plex – missing no edges). It’d have been great if 123 had revealed itself as a clique also, and if 89abc had been detected as a 1plex before 9abc was detected as a clique. How might we make progress in these directions? Try returning to remove all degree ties before moving on? We will try that on the next slide?

  18. K-plex search on G6 continued k-plex=Subgraph missing  k edges. H a kplex and F a ISG(H), then F is a kplex If H is an ISG, |VH|=h, |EH|=H, H=h(h-1)/2, H is a kplex iff H–Hk. If F is missing an edge, H is missing that edge too (K inherits all H edges). F can’t be missing more edges than H. k-core=Subgraph containing  k edges. If F a kcore ISG of H then H is a kcore G6 H0=G={123456789abc} H0=12*11/2=66. H0=19 H0 is a kplex for k  47 deg=333323334434 is a kcore for k19 Mining all kplexes and kcores. At each step, we [potentially] branch to each of the lowest degree vertices (note, I skipped many of them in this illustration.) We might want kplex and/or kcore structure around a particular vertex. Use SP1, SP2…. E.g., find the kplex and kcore structure around v=1: H1=ISG{12346789abc} (deg5=2). H1=11*10/2=55, H1=17. H1 is a kplex for k  37. deg= 33332234434 is a kcore for k17 H26=ISG{1234789abc} (deg6=2). H26=10*9/2=45, H26=15 H26 is a kplex for k  30. deg= 3333124434 is a kcore for k15 H27=ISG{1234689abc} (deg7=2). H27=10*9/2=45 H27=15 H27 is a kplex for k  30. deg= 3332134434is a kcore for k15 (H26 and H27 specify removal of 7 and 6 resp. Thus remove both) SP4 1 2 3 4 5 6 7 8 9 a b c SP1 1 2 3 4 5 6 7 8 9 a b c SP2 1 2 3 4 5 6 7 8 9 a b c SP3 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 3 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 1 1 0 0 1 1 1 0 2 0 0 0 0 0 0 1 0 0 0 0 1 2 1 0 1 1 0 0 0 0 0 0 0 0 3 2 0 0 0 0 0 0 0 1 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 3 3 0 0 0 1 0 0 0 0 1 1 1 0 3 0 0 0 0 1 1 0 0 0 0 0 0 3 0 0 0 0 0 0 1 1 0 0 0 0 4 0 0 0 0 0 0 0 0 1 1 1 0 4 1 1 0 0 0 0 1 0 0 0 0 0 3 4 0 0 1 0 1 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 1 5 1 1 0 0 0 0 0 0 1 1 0 0 5 0 0 1 0 0 0 0 0 0 0 1 1 5 0 0 0 1 0 0 0 1 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 2 6 0 0 0 0 1 0 1 1 0 0 0 0 3 6 1 1 0 0 0 0 0 0 0 0 1 1 6 0 0 0 1 0 0 0 0 1 1 0 0 6 0 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 1 1 1 0 0 0 0 0 0 3 7 0 0 1 0 0 0 0 0 1 1 0 0 7 1 1 0 0 0 0 0 1 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 1 1 b 0 0 1 0 0 0 0 1 0 0 0 0 b 1 1 0 0 0 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 3 b 0 0 0 1 1 0 1 0 0 0 0 0 9 1 1 0 0 1 0 1 0 0 0 0 0 9 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 1 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 1 1 1 4 c 0 0 0 0 1 0 1 0 0 0 0 0 c 0 0 0 1 0 1 0 0 0 0 0 0 c 1 1 0 0 0 0 0 1 0 0 0 0 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 1 1 0 0 0 0 0 0 0 0 8 1 1 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 1 0 1 0 0 0 1 1 8 0 0 0 0 0 1 0 0 1 1 0 0 3 a 0 0 0 0 0 0 0 1 1 0 1 1 4 a 1 1 0 0 1 0 1 0 0 0 0 0 a 0 0 0 1 0 0 0 0 0 0 0 0 a 0 0 1 0 0 1 0 0 0 0 0 0 H2=ISG{123489abc} H2=9*8/2=36 H2=14 H2 is a kplex for k  22. deg= 333224434 is a kcore for k14 H34=ISG{12389abcH34=8*7/2=28 H34=12 H34 is a kplex for k  16. deg= 22324434 is a kcore for k12 H38=ISG{12349abc} H38=8*7/2=28 H38=13 H38 is a kplex for k  15. deg= 33324434 is a kcore for k13 H348=ISG{1239abcH348=7*6/2=21 H384=10 H384 is a kplex for k  11. deg= 2233334 is a kcore for k10 H341=ISG{2389abc} ( H341=7*6/2=21 H341=10 H341 is a kplex for k  11. deg= 1224434 is a kcore for k10 SPL1(1)=234 SPL2(1)=7c SPL3(1)=569abc SPL4(1)=8 To check 1234 kplex/core status check if there are edges, 23 24 34 (y,y,n). Thus, 123, 124 are 0plexes and 3cores. 134, 234 are 1plexes and 2cores. 1234 is a 1plex and a 5core. H342=ISG{1389abc} H342=7*6/2=21 H342=10 H342 is a kplex for k  11. deg= 1224434 is a kcore for k10 (H341,H342,H38 specify removal of 1,2. Thus remove both) H4=ISG{389abcH4=15 H4=9 H4 is a kplex for k  6. deg= 124434 is a kcore for k9 H5=ISG{89abcH5=5*4/2=10 H5=8 H5 is a kplex for k  2. deg= 24433 is a kcore for k8 H6=ISG{9abc} (deg7=2) H6=6 H6=6 H6 is a kplex for k  0. deg= 3333 is a kcore for k6 This is what we want. 89abc a 2plex;9abc a 0plex H0=G={1234567} H=21 H=9 H is a kplex for k  11. deg=3323223 is a kcore for k9 H03=G={124567} H=15 H=8 H is a kplex for k  7. deg=333223 is a kcore for k8 H05=G={123467} H=15 H=8 H is a kplex for k  7. deg=332323 is a kcore for k8 To check 12347c kplex/core status, check edges 17 1c 27 2c 37 3c 47 4c 7c (n n n n n y y n n) 12347c=(Comb(6,2)-7)plex=8plex, 7core H06=G={123457} H=15 H=8 H is a kplex for k  7. deg=332323 is a kcore for k8 1 5 H035=G={12467} H=10 H=8 H is a kplex for k  7. deg= 22312 is a kcore for k8 4 2 6 7 3 H036=G={12457} H=10 H=8 H is a kplex for k  7. deg= 22322 is a kcore for k8 c 9 b H0356=G={1247} H=6 H=4 H is a kplex for k  2. deg= 2231 is a kcore for k4 8 a H03567=G={124} H=3 H=3 H is a kplex for k  0. deg= 222is a kcore for k3 This is what we want. Remove 12489abc H7={3567} H7=6. H7=3H7 is a kplex for k  3. deg=0222 is a kcore for k3 H7={567} H7=3. H7=3 H7 is a kplex for k  0. deg=222 is a kcore for k3

  19. K-Degree-Difference Community Search on G6: A kDegreeDifference Community of a graph, G, is a subgraph, H, such that ddHIntDegH-ExtDegH  k. Theorem: If hH, ddH-h = ddH – (2idh - edh). So we want to remove h s.t. (2idh – edh) is minimum. G6 H= { 35678} id= 02321 ed= 30012 ddH=2 ddH/|VH| = 2/5 = 0.4 2id-ed=-34630 Remove 3 H=G={123456789abc} id= 333323334434 ed= 000000000000 ddH=38 ddH/|VH| = 38/12 = 3.16 Remove 5 H= {12346789abc} id= 33333334434 ed= 00001100000 ddH=34 ddH/|VH| = 34/11 = 3.09 2id-ed=66665568868 Remove 6,7 SP1 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 0 0 0 0 0 0 3 3 1 1 0 0 0 0 0 0 0 0 0 1 3 4 1 1 0 0 0 0 1 0 0 0 0 0 3 5 0 0 0 0 0 1 1 0 0 0 0 0 2 6 0 0 0 0 1 0 1 1 0 0 0 0 3 7 0 0 0 1 1 1 0 0 0 0 0 0 3 b 0 0 0 0 0 0 0 0 1 1 0 1 3 9 0 0 0 0 0 0 0 1 0 1 1 1 4 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 0 0 0 1 0 0 1 1 0 0 3 a 0 0 0 0 0 0 0 1 1 0 1 1 4 H= { 5678} id= 2321 ed= 0012 ddH=5 ddH/|VH| = 5/4 = 1.2 2id-ed= 4630 Remove 8 H= {123489abc} id= 333224434 ed= 000110000 ddH=26 ddH/|VH| = 26/9 = 2.88 2id-ed=666338868 Remove 4,8 H= { 567} id= 222 ed= 011 ddH=4 ddH/|VH| = 4/3 = 1.33 2id-ed= 433 Clique, so remove 567 and start over with 38 (but it has 0 id) H= {1239abc} id= 2233334 ed= 1101100 ddH=16 ddH/|VH| = 16/7 = 2.28 2id-ed=3365568 Remove 1,2 H= {39abc} id= 13334 ed= 21100 ddH=10 ddH/|VH| = 10/5 = 2.0 2id-ed=05568 Remove 3 H= {9abc} id= 3333 ed= 1101 ddH=9 ddH/|VH| = 9/4 = 2.25 2id-ed=5565 Clique so start over with 12345678 H= {12345678} id= 33232331 ed= 00100002 ddH=17 ddH/|VH| = 17/8 = 2.13 2id-ed=66563660 Remove 8 H= {1234567} id= 3323223 ed= 0010010 ddH=16 ddH/|VH| = 16/7 = 2.28 2id-ed=6636436 Remove 3,6 1 5 4 2 6 7 H= {12457} id= 22312 ed= 11011 ddH=6 ddH/|VH| =6/5 = 1.2 2id-ed=33613 Remove 5 3 c 9 b 8 a H= {1247} id= 2231 ed= 1102 ddH=4 ddH/|VH| = 4/4 = 1.0 2id-ed=3360 Remove 7 H= {124} id= 222 ed= 111 ddH=3 ddH/|VH| = 3/3 = 1.0 2id-ed=333 Clique, so start over with 35678

  20. G6 Very Simple Weighted SP1 and SP2 K-plex Search on G6 Weighting: 0,1path nbrs of x times 3; 2path nbrs of x times 2; Until all degrees are weighted, then back to actual subgraph degrees H={123456789abc deg999923634438 x=1 H={123456789abc H=15 H=7 kplex k8 deg999923634438 x=1 after cutting 2,3,4 H={123456789abcH=6 H=5 kplex k1 deg999923634438 x=1, after cut 23468 H={123456789abc H=15 H=7 kplex k8 deg999923634438 x=2 after cutting 2,3,4 H={123456789abcH=6 H=5 kplex k1 deg999923634438 x=2, after cut 23468 H={123456789abc deg999923634438 x=2 H={123456789abcH=3 H=3 0plex deg222623338861 x=3 after cut 1 (actual subgraph degrees) H={123456789abc deg99962333886c x=3 H={123456789abcH=6 H=4 2plex deg99962333886c x=3, after cut 2368 SP2 1 2 3 4 5 6 7 8 9 a b c SP1 1 2 3 4 5 6 7 8 9 a b c SP4 1 2 3 4 5 6 7 8 9 a b c SP3 1 2 3 4 5 6 7 8 9 a b c 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 0 0 0 0 0 0 3 2 0 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 1 1 0 0 1 1 1 0 2 0 0 0 0 0 0 1 0 0 0 0 1 3 0 0 0 0 0 0 1 1 0 0 0 0 3 0 0 0 0 1 1 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 3 3 0 0 0 1 0 0 0 0 1 1 1 0 4 0 0 0 0 0 0 0 0 1 1 1 0 4 1 1 0 0 0 0 1 0 0 0 0 0 3 4 0 0 1 0 1 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 1 5 0 0 0 0 0 1 1 0 0 0 0 0 2 5 1 1 0 0 0 0 0 0 1 1 0 0 5 0 0 1 0 0 0 0 0 0 0 1 1 5 0 0 0 1 0 0 0 1 0 0 0 0 6 0 0 0 0 1 0 1 1 0 0 0 0 3 6 1 1 0 0 0 0 0 0 0 0 1 1 6 0 0 1 0 0 0 0 0 0 0 0 0 6 0 0 0 1 0 0 0 0 1 1 0 0 7 0 0 1 0 0 0 0 0 1 1 0 0 7 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 1 1 1 0 0 0 0 0 0 3 7 1 1 0 0 0 0 0 1 0 0 0 0 b 1 1 0 0 0 1 0 0 0 0 0 0 b 0 0 1 0 0 0 0 1 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 3 b 0 0 0 1 1 0 1 0 0 0 0 0 9 0 0 0 0 0 0 0 1 0 1 1 1 4 9 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 1 0 0 1 0 0 0 0 0 0 9 1 1 0 0 1 0 1 0 0 0 0 0 c 0 0 0 1 0 1 0 0 0 0 0 0 c 0 0 0 0 1 0 1 0 0 0 0 0 c 0 0 1 0 0 0 0 0 1 1 1 0 4 c 1 1 0 0 0 0 0 1 0 0 0 0 8 0 0 0 0 0 1 0 0 1 1 0 0 3 8 1 1 0 0 0 0 0 0 0 0 0 0 8 0 0 1 1 0 0 0 0 0 0 0 0 8 0 0 0 0 1 0 1 0 0 0 1 1 a 0 0 0 0 0 0 0 1 1 0 1 1 4 a 0 0 1 0 0 1 0 0 0 0 0 0 a 1 1 0 0 1 0 1 0 0 0 0 0 a 0 0 0 1 0 0 0 0 0 0 0 0 H={123456789abc deg996946334434 x=4 H={123456789abcH=3 H=3 0plex deg996946334434 x=4 after cut 2346 UNWEIGHTED Degrees H={123456789abc deg333323334434 H={123456789abcH=10 H=5 5plex deg333669964434 x=5 after cut 34 H={123456789abc deg333669964434 x=5 H={123456789abcH=3 H=3 0plex deg333123314434x=5 after cut 1 from SG degs H={123456789abcH=3 H=2 1plex deg33312333223 x=6 after cut 12 SG degs211 H={123456789abc deg333669998834 x=6 H={123456789abc deg333669998834 x=6 after cut 34 H={123456789abcH=3 H=3 0plex deg333122232234x=7 after cut 1 SG degs H={123456789abc deg333969998834x=7 after cut 34 H={123456789abc deg333969934434 x=7 H={123456789abc 2plex deg333342134433 x=8 after cut12 SG degs H={123456789abc deg33334969cc68 x=8 H={123456789abc deg33334969cc68 x=8 after cut 34 H={123456789abcH=10 H=8 H a kplex k 2 deg33632639cc9c x=9 after Cutting 2,3,6 H={123456789abc deg33632639cc9c x=9 H={123456789abcH=10 H=8 H a kplex k 2 deg33632639cc9c x=a after cut 2,3,6 H={123456789abc deg33632639cc9c x=a H={123456789abcH=6 H=6 H a kplex k 0 deg33632639cc9c x=b after cut 2,3,6 H={123456789abc deg33632336cc9c x=b 1 5 H={123456789abcH=6 H=6 H a kplex k 0 deg66932336cc9c x=c after cut 2,3,6 H={123456789abc deg66932336ccpc x=c 4 2 6 7 3 c By weighting the initial round we have gotten nearly perfect information for this example (G6). The weightings, 3 and 2, were arbitrarily chosen but worked here. In general, one should devise a formula to determine them. Also we could weight SP3 and etc. as well? If we have paid the price of constructing SPk k>1, this is a much simpler way to do it, as compared to the Clique Percolation method of Palla (next slide). 9 b 8 a

  21. Very Simple Weighted SP1 k-plex Search on G7 Weighting: 0,1path nbrs of x times 1; 2path nbrs of x times 0; 1 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 1 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 3 2 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 3 2 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 2 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 4 2 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 4 3 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 6 3 3 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 3 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 6 SP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 6 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 9 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 2 3 H=1234567890123456789012345678901234 H=561 H=77 kplx k484 D g9a63444523125222223222533243446bg kcore k77 Cut 123: 1 2 3 H=1234567890123456789012345678901234 H=120 H=38 kplx k82 D 9685322452322522222322243323334367 kcore k38 Cut 23: 1 2 3 H=1234567890123456789012345678901234 H=55 H=26 kplx k24 D 6675322452322522222322223323334344 kcore k26 Cut 24: 1 2 3 H=1234567890123456789012345678901234H=15 H=12 kplx k3 D 5454322422322422222322223323334344kcore k12 G7 Cut 2: 1 2 3 H=1234567890123456789012345678901234H=10 H=10 kplx k0 D 4444322422322422222322223323334344kcore k10 {1,2,3,4, 14} is a clique. {1,2,3,4,9,14} is a 3plex. 2 3 4 4 4 5 5 6 6 6 6 7 8 9 10 11 12 13 15 5 5 1 4 8 2 6 1 3 6 8 0 5 7 9 1 3 5 8 0 2 4 9 2 5 7 1 4 8 2 8 9 5 Cut0: 1 2 3 H=5678901235678901235678901 H=21 H=4 kplx k17 D 2330102000020000002111011 kcore k4 Cut 1 leaves 25 only. 1 2 3 H=56789012356789012345678901234 D 232031200222021202533232435af 1 2 3 H=89023568901235678901 H=19 H=4 kplex k15 D 01000000000002010011 kcore k4 Cut012:1 2 3 H=56789012356789012345678901234 H=55 H=19 kplx k36 D 20203120022202120253323233456 kcore k19 Cut03: 1 2 3 H=56789012356789012345678901234 H=6 H=4 kplx k2 D 20203120022202120223323233222 kcore k6 {24,32,33,34} is a 2plex Cut0: 2 3 H=89023568901235678901 H=19 H=4 kplex k15 D 01000000000002010001 kcore k4 Cut 0 leaves {9,31} as a 0plex 1 2 3 H=5678901235678901235678901 D 2330102000020000002111011 1 2 3 H=89023568901235678901H=17 H=2 kplex k15 D 01000000000002010011 kcore k2 Cut 0 leaves {27,30} as a 0plex Cut01: 1 2 3 H=5678901235678901235678901 H=15 H=6 kplx k9 D 2330102000020000000111011kcore k6 1 2 3 H=89023568901235678901H=14 H=0 kplex k14 D 0100000000000201001kcore k0 no edges left Cut0: 1 2 3 H=5678901235678901235678901 H=10 H=6 kplx k4 D 2330102000020000000111011kcore k6 {5,6,7,11,17} is a 4plex 1 2 3 H=89023568901235678901 D 01000000000002111011 The expected communities are mostly not detected as kplexes or kcores. Cut0: 1 2 3 H=5678901235678901235678901 H=21 H=4 kplx k17 D 2330102000020000002111011 kcore k4 1 2 3 4 5 6 01234567890123456789012345678901234567890123456789012345678901234 0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ@#$ (Symbols for base 65 )

  22. ISG EdgeCount kplex Search Alg on G8 G8 is a graph of word associations starting from the word, BRIGHT using USF Free Association. An edge, AB, means some people associate the word B to word A. We try to determine the 4 categories; Intelligence, Astronomy, Light, Colors . 1 5 4 2 3 41 46 42 8 45 47 1 2 3 4 5 H = 123456789012345678901234567890123456789012345678901234 H=1431 H=197 kplex k1234 Deg 44444bb5656h9747c3c864fag4a386e4546534685768353534965j kcore k197 7 44 6 43 40 9 39 38 53 Cut0-9 1 2 3 4 5 H = 123456789012345678901234567890123456789012345678901234 H=45 H=22 kplex k13 Deg 444442456565974733386446544386545465346857683535349656 kcore k22 48 12 52 10 13 14 11 17 Cut234 1 2 3 4 5 H = 123456789012345678901234567890123456789012345678901234 H=10 H=8 kplex k2 Deg444442456562974733286444344386345465346857683535349654 kcore k8 So {12,24,25,31,54}={sun,yellow,color,red,bright} is a 2plex 36 54 16 24 35 15 23 22 37 21 Attempt 2: Remove bright, double the weight of nbrs of 12 (vertex if max degree) 49 19 34 1 2 3 4 5 H = 12345678901234567890123456789012345678901234567890123 H=1431 H=197 kplx k1234 44444ba5645g9746b2b864f9f49386d4545423675767353534965 20 27 25 18 50 Cut 1-9 1 2 3 4 5 H = 12345678901234567890123456789012345678901234567890123H=1431 H=197 kplex k1234 44484mka68agie4cm2b8c4fif49386d454542367576e356a349c5 51 26 G8 30 29 28 Cut 3 1 2 3 4 5 H = 1234567890123456789012345678901234567890123456789012 H=1431 H=197 kplex k1234 44484664684c66467238444444938634545423675764356334935 33 31 32 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 6 1 2 0 0 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 7 1 3 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 9 1 4 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 7 1 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 6 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 7 1 7 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 2 1 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 2 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 4 2 3 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 5 2 4 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 6 2 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 2 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 5 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 4 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 5 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 4 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 6 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 5 3 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 3 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 3 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 6 4 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 8 4 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 4 2 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 7 4 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 6 4 4 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 8 4 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 3 4 6 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 4 7 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 4 8 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 4 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 4 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 9 5 2 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 5 3 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 4 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 9 SP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 4 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 4 3 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 4 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 5 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 6 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 7 0 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 8 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 5 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 6 1 Scientist 2 Science 3 Astronomy 4 Earth 5 Space 6 Moon 7 Star 8 Ray 9 Intelligent 10 Golden 11 Glare 12 Sun 13 Sky 14 Moonlight 15 Eyes 16 Sunshine 17 Light 18 Lit 19 Dark 20 Brown 21 Tan 22 Orange 23 Blue 24 Yellow 25 Color 26 Gray 27 Black 28 Race 29 White 30 Green 31 Red 32 Crayon 33 Pink 34 Velvet 35 Flashlight 36 Glow 37 Dim 38 Gifted 39 Genius 40 Smart 41 Inventor 42 Einstein 43 Brilliant 44 Shine 45 Laser 46 Telescope 47 Horizon 48 Sunset 49 Ribbon 50 Violet 51 Purple 52 Beam 53 Night 54 Bright

  23. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 3 4 5 6 7 8 9 50 1 2 3 SP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 SP1 and SP2 for G8 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 SP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 3 4 5 6 7 8 9 50 1 2 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 4 4 4 b a 5 6 4 5 g 9 7 4 6 b 2 b 8 6 4 f 9 f 4 9 3 8 6 d 4 5 4 5 4 2 3 6 7 5 7 6 7 3 5 3 5 3 4 9 6 5

  24. Simple Weighted SP1, SP2 K-plex Search on G8 1 5 4 2 Weighting 444 0,1path neighbors (12012) times 5 334 2 path nbrs (39893) times 3 3 41 46 42 8 45 47 7 44 6 43 40 9 39 38 53 48 12 52 10 13 14 11 17 36 This gives C0={1,2,9,39,40,41,42,43} which is exactly the Intelligence Class except that v=38 (gifted) is missing. It is a kplex k8 (not that strong of a community!) 54 16 24 35 15 23 22 37 21 49 19 34 Within the Intelligence Class this is the 1plex, C1={1, 2,40,41,42} ( only edge missing is (2,40) ) with C1-degrees: 4 3 3 4 4 Thus if we cut next using C1-degrees (cut 2,40) leaves the clique (0plex) C2={1,41,42} Cutting C0 and starting over: 20 27 25 18 50 11 1 1 1 1 1 1 44444105645697461218645954938634545423675767353534965 12345678901234567890123456789012345678901234567890123 51 26 121663 23954136 3231353 2 212 1114 113 131 44242600640642266634620404734864545223675782958094865 cut<30 G8 30 29 28 Weighting 0,1path neighbors (367) times 5 1111445 2 path nbrs (452347483) times 3 33 31 32 This gives C2={3,4,5,6,7, 12,13,14,15,17,23,25,31,44, 48, 53} Whereas, Astronomy is 3,4,5,6,7,8,10,11,12,13,14,16,17, 44,45,46,47,48,52,53 so, not a good fit! Weighting 0,1 SP nbrs times 6 With replacement but using as starting vertex, the remaining vertex of highest degree (first, v=12). Weighting 0,1 SP nbrs times 5 2 SP nbrs times 3 221 11 1 1 1 1 1 1 1 13231 00244105845697461218645954938634545429855587353534965 next cut<18 11 1 1 1 1 1 1 44444105645697461218645954938634545423675767353534965 11 1 1 1 1 1 1 44444105645697461218645954938634545423675767353534965 12155222143135323144122131113112231 44202505605655205634025554734894545823675785955594705 cut<20 221 11 1 1 1 1 1 1 1 13231 00244105845697461218645954938634545429855587353534965 instead cut<19 221 11 1 1 1 1 1 1 1 13231 00244105845697461218645954938634545429855587353534965 11 1 2 1 1439512514372325 44444105645697461218640454488684045423675767353534465 12155222143135323144122131113112231 44202505605655205634025554734894545823675785955594705 cut<20 21155 14223 1 1 1 1 111 44522505645887163218645954938634545421675768353534965 next cut<10 21155 142231 1 1 1 111 44522505645887163218645954938634545421675768353534965 next cut<12 11 1 1 1 1 1 1 44544105645697461218645954938634545421675766353534965 G-C0 degs 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 12345678901234567890123456789012345678901234567890123 x=25 12345678901234567890123456789012345678901234567890123 x=12 12345678901234567890123456789012345678901234567890123 x=3 12345678901234567890123456789012345678901234567890123 x=3 12345678901234567890123456789012345678901234567890123 x=3 12345678901234567890123456789012345678901234567890123 x=12 11111 11 44444 55 Astronomy is 345678 01234 67 45678 23 Weighting 0,1 SP nbrs times 6 2 SP nbrs times 3 Astronomy is 345678 01234 67 45678 23 1234567890123456789012345678901234567890123456789012 5 astronomy vertices missing (3,5,45,46,53} and 2 non-astronomy included {21,24} Weighting 0,1 SP nbrs times 6 2 SP nbrs times 1 Colors is 5 012345678901234 901 4 colors missing but zero non-colors included. 44444ba5645g9746b2b864f9f49386d4545423675767353534965

  25. While constructing Shortest Path pTrees, SP2…, record the Shortest Path Participation Count of each edge (SPPC). The edge(s) with max SPPC should be the best candidates for removal? G5 2 1 1 2 2 ct 3 2 2 1 2 2 2 2 ct 3 2 2 1 2 2 2 2 ct 2 1 1 2 2 ct 12 10 2 4 4 2 4 2 ct 0 0 0 0 0 0 0 0 ct 1 2 0 1 1 0 1 0 ct 0 0 0 2 1 0 1 0 ct 0 0 0 0 0 ct 1 2 3 4 5 6 7 8 SPPC) 1 2 4 5 7 SP2 1 2 3 4 5 6 7 8 E 1 2 3 4 5 6 7 8 SP2 1 2 3 4 5 6 7 8 SP3 1 2 4 5 7 SP 1 2 3 4 5 6 7 8 SP 1 2 3 4 5 6 7 8 SP4 1 2 4 5 7 E 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 6 0 0 3 0 3 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 6 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 2 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 Delete (1,2) And {3,6,8} and do over. 4 3 3 3 1 3 1 ct Delete (1,6) and do over. 1 2 3 4 5 6 7 E 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 2 3 4 5 6 7 3 3 3 3 1 2 1 ct 1 2 3 4 5 6 7 E 0 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 2 3 4 5 6 7 2 1 1 1 2 3 2 ct 0 0 0 0 1 0 1 ct 1 2 3 4 5 6 7 SP2 1 2 3 4 5 6 7 SP2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 8 6 G2 5 4 7 0 2 2 2 3 0 3 ct 0 0 0 0 0 0 0 ct 7 SP gives connectivity comp partition: CC(1)={1,5,7} is a 0plex since EdgeCt=3=COMBO(3,2)-0. CC(2)={2,4} is a 0plex since EdgeCt=1=COMBO(2,2)-0. 1 2 3 4 5 6 7 SP3 1 2 3 4 5 6 7 SP3 5 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 1 4 3 3 3 1 3 1 ct 3 3 3 3 2 2 2 ct 1 2 3 4 5 6 7 SP=SP1 | SP2 | SP3 1 2 3 4 5 6 7 SP SP gives connectivity comp partition: CC(1)={1,2,4,5,7} is a 5plex since EdgeCt=5=COMBO(5,2)-5. CC(3)={3,6,8} is a 0plex since EdgeCt=3=COMBO(3,2)-0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 2 3 4 5 6 7 4 3 3 3 3 3 2 2 2 ct 24 6 6 6 5 22 5 ct 1 2 3 4 5 6 7 SPPC 1 2 3 4 5 6 7 SPPC 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 4 0 1 1 0 0 0 4 1 0 1 0 0 0 4 1 1 0 0 0 0 0 0 0 0 0 5 0 c 0 0 0 5 0 5 0 0 0 0 0 5 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 0 1 1 1 0 0 0 0 4 4 4 0 c 0 SP gives connectivity comp partition: CC(1) = {1}List(SP(1) = {1,2,3,4,5,6,7} is a 12plex since EdgeCt=9=COMBO(7,2)-12 SP gives the connectivity component partition: CC(1)={1,2,3,4} 0plex since EdgeCt=12= 2*COMBO(4,2) CC(5)={5,6,7} 1plex since EdgeCt=4=2*(COMBO(3,2)-1)

  26. 3 2 P 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 G1 SPPC 0 4 0 0 0 4 0 4 4 4 0 4 0 0 0 0 4 0 0 0 0 4 0 0 0 2Pkey 1,1,1 1,1,2 1,1,3 1,1,4 1,2,1 1,2,2 1,2,3 1,2,4 1,3,1 1,3,2 1,3,3 1,3,4 1,4,1 1,4,2 1,4,3 1,4,4 2,1,1 2,1,2 2,1,3 2,1,4 2,2,1 2,2,2 2,2,3 2,2,4 2,3,1 2,3,2 2,3,3 2,3,4 2,4,1 2,4,2 2,4,3 2,4,4 3,1,1 3,1,2 3,1,3 3,1,4 3,2,1 3,2,2 3,2,3 3,2,4 3,3,1 3,3,2 3,3,3 3,3,4 3,4,1 3,4,2 3,4,3 3,4,4 4,1,1 4,1,2 4,1,3 4,1,4 4,2,1 4,2,2 4,2,3 4,2,4 4,3,1 4,3,2 4,3,3 4,3,4 4,4,1 4,4,2 4,4,3 4,4,4 GN: Compute all edge betweenesses (SPPCs) Remove edge with largest betweeness Recalc betweenesses; Repeat. 2 2 4 4 4 4 4 4 SPPC 0 5 0 0 0 5 0 5 4 5 0 5 0 4 0 0 4 4 0 4 0 5 0 4 0 3 3 2 2 3 3 3 3 3 2 2 2 2 2 2 1 1 1 1 G1_1 1 1 1 1 1 1 G1_2 G1_2 G1_3 G1_3 G1_3 G1_3 G1_3 G1_3 4 4 5 5 5 5 5 5 5 5 null S 1 P 1 0 1 0 0 0 1 S 1 P 1 0 1 0 0 0 1 S 1 P 5 0 1 0 0 0 1 S 1 P 5 0 1 0 1 0 2 S 1 P 2 1 0 1 0 1 3 S 1 P 2 1 0 1 1 1 4 S 1 P 3 0 1 0 0 0 1 S 1 P 3 0 1 0 1 0 2 S 1 P 4 0 1 0 0 0 1 S 1 P 4 0 0 1 0 1 2 SPPC 0 0 3 2 0 0 0 3 3 0 0 4 1 3 4 0 null S 1 P 1 0 0 1 0 1 S 1 P 1 0 0 1 1 2 S 1 P 2 0 0 0 1 1 S 1 P 2 0 0 0 1 1 S 1 P 3 1 0 0 1 2 S 1 P 3 1 0 0 1 2 S 1 P 4 0 1 1 0 2 S 1 P 4 1 1 1 0 3 3 3 4 4 null null nul SPPC 0 0 1 2 0 0 0 3 1 0 0 2 2 3 2 0 null E 0 1 0 0 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 E 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 E 0 1 0 0 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 E 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 E 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 Ekey 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4,3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 Ekey 1,1 1,2 1,3 1,4 1,5 2,1 2,2 2,3 2,4 2,5 3,1 3,2 3,3 3,4 3,5 4,1 4,2 4,3 4,4 4,5 5,1 5,2 5,3 5,4 5,5 Ekey 1,1 1,2 1,3 1,4 1,5 2,1 2,2 2,3 2,4 2,5 3,1 3,2 3,3 3,4 3,5 4,1 4,2 4,3 4,4 4,5 5,1 5,2 5,3 5,4 5,5 Ekey 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 Ekey 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 null null nul nul nul S 2 P 1 3 0 0 0 1 1 S 2 P 2 4 0 0 1 0 133 S 2 P 3 4 0 1 0 0 1 S 2 P 4 3 1 0 0 0 1 S 2 P 5 2 1 0 1 1 0 3 S 2 P 1 2 0 0 1 1 1 3 S 2 P 3 2 1 0 0 1 1 3 S 2 P 4 2 1 0 1 0 1 3 S 2 P 3 2 1 0 0 0 1 2 S 2 P 3 4 0 0 0 0 1 1 S 2 P 4 3 0 1 0 0 0 1 S 2 P 4 5 0 1 0 0 0 1 S 2 P 5 2 1 0 1 0 0 2 S 2 P 5 4 0 0 1 0 0 1 S 2 P 1 2 0 0 1 0 1 2 S 2 P 2 3 0 0 0 1 0 1 S 2 P 2 5 0 0 0 1 0 1 null Check SPPC(34)=SPPC(43) (verify SPs backwards from hk get counted.) (34)E so ct=1 + CountS2P(34)=1 + CountS2P(43)=1 so ct=3 + CtS3P(34g)=0 + CtS3P(g34)=1, g=1 ct=4 GN says delete (3,4)! nul GN says delete any edge! nul nul nul S 2 P 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 P 1 4 0 1 0 0 1 S 2 P 2 4 1 0 1 0 2 S 2 P 3 4 0 1 0 0 1 nul nul nul nul nul nul nul nul nul S 3 P 1 3 4 0 1 0 0 S 3 P 2 4 3 1 0 0 0 null null null S 3 P 1 2 3 0 0 0 1 0 1 S 3 P 1 2 5 0 0 0 1 0 1 S 3 P 4 3 2 1 0 0 0 0 1 S 3 P 4 5 2 1 0 0 0 0 1 nul GN says delete 12 | 25 | 34 | 36 null null To construct SPPC(hk) =SPPC(kh) (Shortest Path Participation Count) if (hk)E count 1 + OneCountS2P(hk) + OneCountS2P(kh) + OneCountS3P(hkg) + OneCountS3P(ghk), g + OneCountS4P(hkfm) + OneCountS4P(fhkm) + OneCountS4P(fmhk) f,m. Etc. GN: delete 12 | 23 | 25 not 34, 45 6 6 6 6 6 6 6 6 S 1 P 1 0 1 0 0 0 1 2 S 1 P 5 0 1 0 1 0 0 2 S 1 P 6 1 0 1 0 0 0 2 S 1 P 2 1 0 1 0 1 0 3 S 1 P 3 0 1 0 1 0 1 3 S 1 P 4 0 0 1 0 1 0 2 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 G1_4 G1_4 G1_4 G1_4 G1_4 G1_4 G1_4 G1_4 5 5 5 5 5 5 5 5 not 23, 16, 45 SPPC 0 7 0 0 5 7 0 6 4 7 0 6 0 7 0 7 0 4 7 0 5 0 7 0 5 5 7 0 S 2 P 1 6 0 0 1 0 0 0 1 S 2 P 2 1 0 0 0 0 0 1 1 S 2 P 3 2 1 0 0 0 1 0 2 S 2 P 3 4 0 0 0 0 1 0 1 S 2 P 3 6 1 0 0 0 0 0 1 S 2 P 4 3 0 1 0 0 0 1 2 S 2 P 4 5 0 1 0 0 0 0 1 S 2 P 5 2 1 0 1 0 0 0 2 S 2 P 5 4 0 0 1 0 0 0 1 S 2 P 6 1 0 1 0 0 0 0 1 S 2 P 1 2 0 0 1 0 1 0 2 S 2 P 2 3 0 0 0 1 0 1 2 S 2 P 2 5 0 0 0 1 0 0 1 S 2 P 6 3 0 1 0 1 0 0 2 SPPC recalculation and repeat steps? Anyone see a shortcut? Or do we just start the calculation over on the reduced graph? Do the pointers help? Since in S2P(hk) one has to search out S2P(kh) and in S3P(hk) one has to find all S3P(hkg) snf D3P(ghk) g In the appendix I begin work on uniquely representing shortest k paths using both a fore and aft pTree. Consider that in G1_4 S3P(16)=2. • Notes: • If any OneCount=0, no subsequence exist. • It might be useful to use ptrs to make this proc easier. • GN edge betweenness specifies pruning (2,4) S 3 P 1 2 3 0 0 0 1 0 0 1 S 3 P 1 2 5 0 0 0 1 0 0 1 S 3 P 1 6 3 0 0 0 1 0 0 1 S 3 P 4 3 2 1 0 0 0 0 0 1 S 3 P 4 3 6 1 0 0 0 0 0 1 S 3 P 4 5 2 1 0 0 0 0 0 1 S 3 P 6 1 2 0 0 0 0 1 0 1 S 3 P 6 3 2 0 0 0 0 1 0 1 S 3 P 6 3 4 0 0 0 0 1 0 1 S 3 P 5 2 1 0 0 0 0 0 1 1 S 3 P 5 2 3 0 0 0 0 0 1 1 S 3 P 5 4 3 0 0 0 0 0 1 1

  27. 2 1 2 3 2 3 2 1 1 1 1 3 2 0 2 2 E 31 7 31 29 7 27 7 7 3 3 3 9 2 0 2 2 SPPC a e b 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 c 0 0 0 0 0 0 0 0 3 3 3 0 0 0 0 0 8 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 g 0 0 f 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 3 g 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 7 f 0 0 0 0 7 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 6 f 0 0 0 6 0 6 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 6 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 9 d 1 2 3 4 5 6 7 8 9 a b c d e f g 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 f g b c 3 2 3 1 1 1 1 2 2 2 2 0 0 0 0 0 SP2 SP gives connectivity comp partition: CC(1)={1 2 3 4 5 6 7 8} is a 20plex since EdgeCt=8=COMBO(8,2)-20. CC(9)={9 a b c} is a 3plex since EdgeCt=3=COMBO(4,2)-3 CC(d)={d f g} is a 0plex since EdgeCt=3=COMBO(3,2)-0. CC( e)={e} 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 6 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 SP2 all pure0 2 1 2 1 1 1 1 1 0 0 0 0 0 0 0 0 SP3 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 b 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 8 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 3 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 4 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 5 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 6 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 7 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 SP gives connect comps: CC(1)={1}, CC(5)={5 6 7} Is a 0plex since EdgeCt34=COMBO(3,2)-0 Done! b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 7 Delete (1,3) (SPPC=16 max) and delete {d f g}, {e} and do over. Also delete {9 a b c} as a 4VetexHubSpoke3plex. 7 7 7 7 7 7 7 7 3 3 3 3 2 0 2 2 SP 0 1 0 2 1 2 1 1 0 0 0 0 0 0 0 0 SP4 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 1 0 2 2 2 SP 0 2 2 2 E 1 1 1 3 2 3 2 1 E 2 2 2 0 1 0 1 2 SP2 3 3 3 3 3 3 3 3 SP 3 3 3 9 9 5 4 3 SPPC (Shortest Path Participation Counts) SP3 all pure0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 8 0 1 1 1 0 0 0 0 8 0 1 1 0 0 0 0 0 8 0 0 0 1 0 0 0 0 8 0 0 0 3 0 0 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 0 0 0 0 0 3 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 5 0 0 0 0 0 1 1 0 2 0 0 0 3 0 0 0 0 2 0 0 1 1 0 0 0 1 2 0 0 0 1 0 0 0 0 2 0 0 1 0 0 0 0 1 6 1 0 0 0 1 0 1 0 6 1 0 0 0 1 0 1 0 3 0 0 0 3 0 0 0 0 3 0 1 0 1 0 0 0 1 3 0 1 0 0 0 0 0 1 3 0 0 0 1 0 0 0 0 7 0 0 0 0 1 1 0 0 7 0 0 0 0 1 1 0 0 4 0 3 3 0 0 0 0 3 4 0 1 1 0 0 0 0 1 4 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 5 0 0 0 0 0 1 1 0 5 1 0 0 0 0 1 1 0 5 1 0 0 0 0 0 0 0 5 0 0 0 0 0 2 2 0 6 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 6 1 0 0 0 1 0 1 0 6 3 0 0 0 1 0 1 0 7 0 0 0 0 2 2 0 0 7 0 0 0 0 1 1 0 0 7 1 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 SP gives connect comps: CC(1)={1 5 6 7} 2plex EdgeCt=4=COMBO(4,2)-2. CC(2)={2 3 4 8} is a 3plex since Ect=3=COMB(4,2)-3 (a 4VertexHubSpoke) 0 2 0 0 1 0 2 2 0 0 0 0 0 0 0 0 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SP6 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 Delete{2 3 4 8} 4VHubSpoke3plex, (1,6) G6

  28. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 E E 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 SP2 9 13 19 16 14 13 13 17 25 19 14 15 14 25 15 15 3 15 16 26 15 16 16 15 6 6 14 20 21 15 20 26 11 6 SP3 8 11 4 11 8 8 8 11 3 11 8 9 9 3 6 6 12 8 6 4 6 8 6 4 23 23 6 8 8 5 8 1 10 10 SP4 0 0 0 0 8 8 8 1 0 1 8 8 8 0 9 9 8 8 8 0 8 8 8 8 1 1 10 1 1 8 1 0 1 1 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 8 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 wt V#> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 SP1 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 -1 SP2 9 13 19 16 14 13 13 17 25 19 14 15 14 25 15 15 3 15 16 26 -1 SP3 8 11 4 11 8 8 8 11 3 11 8 9 9 3 6 6 12 8 6 4 -1 SP4 0 0 0 0 8 8 8 1 0 1 8 8 8 0 9 9 8 8 8 0 -1 SP5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 8 0 1 0 WeightSum 15 -6 -3 -15 -24 -21 -21 -21 -18 -27 -24 -30 -27 -18 -27 -27 -27 -27 -27 -24 Nbrs1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 Nbrs34 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 -20 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 -20 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 If ( WtSum>=-20 & Nbr(1) ) then 1 else 0. This is an Agglomerative Method based on weighted sum of SPk counts to identify 1 and 34 as centers. Then among their individual nbrs, wt V#> 21 22 23 24 25 26 27 28 29 30 31 32 33 34 2 SP1 2 2 2 5 3 3 2 4 3 4 4 6 11 16 -1 SP2 16 16 16 15 6 6 14 20 21 15 20 26 11 6 -1 SP3 6 8 6 4 23 23 6 8 8 5 8 1 10 10 -1 SP4 8 7 8 8 1 1 10 1 1 8 1 0 1 1 -1 SP5 1 0 1 1 0 0 1 0 0 1 0 0 0 0 WeightSum -27 -27 -27 -18 -24 -24 -27 -21 -24 -21 -21 -15 0 15 Nbrs1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 Nbrs34 1 0 1 1 0 0 1 1 1 1 1 1 0 1 -20 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -20 1 0 1 1 0 0 1 1 1 1 1 1 0 1 select their communities with a threshold on the weighted sum (=-20) giving the light green “1community” and black “34community (overlapping). Next, excise those and iterate. When all are in a community probably do a k means reshuffle to improve? Using weights of 0,1,2,4,6 for SP1,2,3,4,5 resp. wt V#> 5 6 7 8 11 12 13 17 18 22 25 26 33 0 SP1 3 4 4 4 3 1 2 2 2 2 3 3 11 1 SP2 14 13 13 17 14 15 14 3 15 16 6 6 11 2 SP3 8 8 8 11 8 9 9 12 8 8 23 23 10 4 SP4 8 8 8 1 8 8 8 8 8 7 1 1 1 6 SP5 0 0 0 0 0 0 0 8 0 0 0 0 0 WeightSum 62 61 61 43 62 65 64 107 63 60 56 56 35 SP1|2(17) 1 1 1 0 1 0 0 1 0 0 0 0 0 60 1 1 1 0 1 0 0 1 0 0 0 0 0 15,16,19,21,23,24,27,30 only 17 on, 5deg=1 Iterate again on the remaining G7 17 SP5 8=5dg Using weights of5,5,1,1,0 for SP1,2,3,4,5 resp. wt V#> 8 12 13 18 22 25 26 33 5 SP1 4 1 2 2 2 3 3 11 5 SP2 17 15 14 15 16 6 6 11 1 SP3 11 9 9 8 8 23 23 10 1 SP4 1 8 8 8 7 1 1 1 0 SP5 0 0 0 0 0 0 0 0 WeightSum 117 97 97 101 105 69 69 121 SP1|2(8) 1 1 1 1 1 0 0 0 SP1|2(33) 0 0 0 0 0 1 1 1 97 1 1 1 1 1 0 0 0 69 0 0 0 0 0 1 1 1 This method uses site betweeness, not edge betweenenss (SPPC not computed) but gives a good overlapping clustering (close to the author’s). One could attempt a few kMeans rounds to try to improve it. 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 10,25,26,28,29, 31 33,34 not shown (only 17 on, 8 only 27 turned on 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8=4dg 5 6 7 11 2 3 5 6 7 8 9 21 2 3 4 7 30 SP4 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

More Related