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Divisive Algorithms for Community Detection

Learn about divisive algorithms, a simple way to identify communities in a graph by removing inter-community edges, and the popular Girvan and Newman algorithm.

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Divisive Algorithms for Community Detection

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  1. DIVISIVE ALGORITHMS A simple way to identify communities in a graph is to detect the edges that connect vertices of different communities and remove them, so that the clusters get disconnected from each other. This is the philosophy of divisive algorithms. The crucial point is to find a property of intercommunity edges that could allow for their identification. • With divisive hierarchical clustering one removes inter-cluster edges instead of edges between pairs of vertices with low similarity and there is no guarantee, apriori, that inter-cluster edges connect vertices with low similarity. • In some cases vertices (with all their adjacent edges) or whole subgraphs may be removed, instead of single edges. Being hierarchical clustering techniques, it is customary to represent the resulting partitions by means of dendrograms. • The most popular algorithm is that proposed by Girvan and Newman (Girvan and Newman,2002; 2004). The method is historically important, because it marked the beginning of a new era in the fieldof community detection. Edges are selected based on a measures of edge centrality (importance wrt to a property or process running on the graph. • Computation of the centrality for all edges; • Removal of edge with largest centrality: in case of ties with other edges, one is picked at random; • Recalculation of centralities on the running graph; • Iteration of the cycle from step 2. • Girvan and Newman used betweenness, expressing the frequency of the participation of edges to a process. They used 3 alternative definitions: geodesic edge betweenness, random-walk edge and current-flow edge betweenness. • Edge betweennessis the number of shortest paths (geodesics) between all vertex pairs that run along the edge. • It extends site betweenness(Freeman,1977)and expresses the importance of edges in processes like information spreading, where information usually flowsthrough shortest paths. It is intuitive that intercommunity edges have a large value of the edge betweenness, because many shortest paths connecting vertices of different communities will pass through them. • As in the calculation of site betweenness, if there are two or more geodesic paths with the same endpoints that run through an edge, the contribution of each of them to the betweennessof the edge must be divided by the multiplicity of the paths, as one assumes that the signal/information propagates equally along each geodesic path. The betweenness of all edges of the graph can be calculated in time that scales asO(|E|*|V|), or O(|V|2) on a sparse graph, with techniques based on breadth-first search.

  2. In info spreading, signals flow across random rather than geodesic, so edge betweenness is given by the frequency of the passages across the edge of a random walker (random-walk betweenness). A random walker moving from a vertex follows each adjacent edge with equal probability. A pair of vertices is chosen at random, s and t. The walker starts at s and keeps moving until it hits t, where it stops. One computes the probability that each edge was crossed by the walker, and averages over all possible choices for the vertices s and t. It is meaningful to compute the net crossing probability, which is proportional to the number of times the walk crossed the edge in one direction. One neglects back and forth passages that are accidents of the random walk and tell nothing about the centrality of the edge. Calculation of random-walk betweenness requires the inversion of an n n matrix (once), followed by obtaining and averaging the flows for all pairs of nodes. The first task requires a time O(n3), the second O(mn2), for a total complexity O[(m+n)n2], or O(n3) for a sparse matrix. The complete calculation requires a time O(n3) on a sparse graph. Current- flow betweenness is defined by considering the graph a resistor network, with edges having unit resistance. If a voltage difference is applied between 2 vertices, each edge carries some amount of current, that can be calculated by solving Kircho's equations. The procedure is repeated for all possible vertex pairs: the current-flow betweenness of an edge is the average value of the current carried by the edge. It is possible to show that this measure is equivalent to random-walk betweenness, as the voltage differences and the random walks net flows across the edges satisfy the same equations (Newman,2005). Therefore, the calculation of current-flow betweenness has the same complexity. Calculating edge betweenness is faster than current-flow or random walk [O(n2) versus O(n3) on sparse graphs]. In practice GN-edge betweenness gives better results. Tyler et al. proposed a modification to improve the speed of the calculation (Tyler et al.,2003;Wilkinson and Huberman,2004). The gain in speed was required for graphs of gene co-occurrences (gene-gene interactioin graph), which are too large for Girvan and Newman. Algorithms computing site/edge betweenness start from any vertex as center and compute the betweenness from all paths originating at that vertex; the procedure is then repeated for all vertices (Brandes01;GN04;Zhouet,06). Tyler et al. proposed to calculate the contribution to edge betweenness only from a limited number of centers, chosen at random, deriving a sort of Monte Carlo estimate. Numerical tests indicate that, for each connected subgraph, it sufices to pick a number of centers growing as the logarithm of the number of vertices of the component. For a given choice of the centers, the algorithm proceeds just like GN but the stopping condition does not require resulting partition modularity calculation, but relies on a particular definition of community, a connected subgraph with n0 vertices is a community if the edge betweenness of any of its edges does not exceed n0-1. Indeed, if the subgraph has 2 parts connected by a single edge, the betweenness of the edge is n0-1 with equality holding only if one of the two parts consists of a single vertex. Therefore, the condition on the betweenness of the edges would exclude such situations, although other types of cluster structures might still be compatible with it. Edges are removed until all connected components of the partition are communities. Monte Carlo sampling of edge betweenness induces statistical errors. As a consequence the partitions are in general different for different choices of the set of center vertices. However by repeating the calculation many times, the method gives good results on a network of gene co-occurrences (Wilkinson04), with a substantial gain of computer time. The technique has been also applied to a network of people corresponding via email (Tyler et al.,2003). In practical examples, only vertices lying at the boundary between communities may not be clearly classified, and be assigned sometimes to a group, sometimes to another. This is nice since it identifies overlaps between communities, as well as the degree of membership of overlapping vertices in the clusters they belong to. GN cannot do this. Another fast version of GN makes a quick approximation of the edge betweenness values by using a network structure index, which consists of a set of vertex annotations combined with a distance measure (Rattigan et al.,2006). Basically one divides the graph into regions and computes the distances of every vertex from each region. In this way Rattigan et al. showed that it is possible to lower the complexity of the algorithm to O(m), by keeping a fair accuracy in the estimate of the edge betweenness values. This version gives good results on the benchmark graphs proposed by Brandes et al. (Brandes et al.,2003), as well as on a collaboration network of actors and on a citation network. Chen and Yuan have pointed out that counting all possible shortest paths in the calculation of the edge betweenness may lead to unbalanced partitions, with communities of very different size, and proposed to count only non-redundant paths, i. e. paths whose endpoints are all different from each other: the resulting betweenness yields better results than standard edge betweenness for mixed clusters on the benchmark graphs of Girvan andNewman (Chen and Yuan,2006). Holme et al. have used a modified version of the algorithm in which vertices, rather than edges, are removed (Holme et al.,2003). A centrality measure for the vertices, proportional to their site betweenness, and inversely proportional to their in-degree, is chosen to identify boundary vertices, which are then iteratively removed with all their edges. This modification, applied to study the hierarchical organization of biochemical networks, is motivated by the need to account for reaction kinetic information, that simple site betweenness does not include. The indegree of a vertex is solely used because it indicates the number of substrates to a metabolic reaction involving that vertex; for the purpose of clustering the graph is considered undirected, as usual.

  3. The algorithm of Girvan and Newman is unable to find overlapping communities, as each vertex is assigned to a single cluster. Pinney and Westhead have proposed a modification of the algorithm in which vertices can be split between communities To do that, they also compute the betweenness of all vertices of the graph. Unfortunately the values of edge and site betweenness cannot be simply compared, due to their different normalization, but the authors remarked that the two endvertices of an inter-cluster edge should have similar betweenness values, as the shortest paths crossing one of them are likely to reach the other one as well through the edge. So they take the edge with largest betweenness and remove it only if the ratio of the betweenness values of its endvertices is between... Otherwise, the vertex with highest betweenness (with all its adjacent edges) is temporarily removed. When a subgraph is split by vertex or edge removal, all deleted vertices belonging are copied in each subcomponent, along with all their edges. Gregory has a similar approach, named CONGA in which vertices are split among clusters if their site betweenness exceeds the maximum value of the edge betweenness . A vertex is split by assigning some of its edges to one of its duplicates, and the rest to the other. There are several possibilities to do that, Gregory proposed to go for the split that yields the maximum of a new centrality measure, called split betweenness, which is the number of shortest paths that would run between two parts of a vertex if the latter were split. The method has a worst-case complexity O(m3), or O(n3) on a sparse graph. Code is at http://www.cs.bris.ac.uk/~steve/networks/index.html. Another promising track to detect inter-cluster edges is related to the presence of cycles, i. e. closed non-intersecting paths, in the graph. Communities are characterized by a high density of edges, so it is reasonable to expect that such edges form cycles. On the contrary, edges lying between communities will hardly be part of cycles. Based on this intuitive idea, Radicchi et al. proposed a new measure, the edge clustering coefficient, such that low values of the measure are likely to correspond to intercommunity edges. The edge clustering coefficient generalizes to edges the notion of clustering coefficient. The measure is (anti)correlated with edge betweenness: edges with low edge clustering coefficient usually have high betweenness and vice versa, although the correlation is not perfect. The method works as the algorithm by Girvan and Newman. At each iteration, the edge with smallest clustering coefficient is removed, the measure is recalculated again, and so on. If the removal of an edge leads to a split of a subgraph in two parts, the split is accepted only if both clusters are LS-sets (“strong") or “weak" communities. The verification of the community condition on the clusters is performed on the full adjacency matrix of the initial graph. If the condition were satisfied only for one of the two clusters, the initial subgraph may be a random graph, as it can be easily seen that by cutting a random graph in two parts, the larger of them is a strong (or weak) community with very high probability, whereas the smaller part is not. Enforcing the community condition on both clusters, it is more likely that the subgraph to be split indeed has a cluster structure. Therefore, the algorithm stops when all clusters produced by the edge removals are communities in the strong or weak sense, and further splits would violate this condition. The authors suggested to use the same stopping criterion for the algorithm of Girvan and Newman, to get structurally well-defined clusters. Since the edge clustering coefficient is a local measure, involving at most an extended neighborhood of the edge, it can be calculated very quickly. The running time of the algorithm to completion is O(m4=n2), or O(n2) on a sparse graph, if g is small, so it is much shorter than the running time of the Girvan-Newman method. The software of the algorithm can be found in http://filrad.homelinux.org/Data/. The algorithm may give poor results when the graph has few cycles, as it happens in some social and many non- social networks. In this case, in fact, the edge clustering coefficient is small and fairly similar for most edges, and the algorithm may fail to identify the bridges between communities. An alternative measure of centrality for edges is information centrality. It is based on the concept of efficiency ( Latora and Marchiori,2001), which estimates how easily information travels on a graph according to the length of shortest paths between vertices. The efficiency of a network is defined as the average of the inverse distances between all pairs of vertices. If the vertices are “close" to each other, the efficiency is high. The information centrality of an edge is the relative variation of the efficiency of the graph if the edge is removed. In the algorithm by Fortunato et al. (Fortunato et al.,2004), edges are removed according to decreasing values of information centrality. The method is analogous to that of Girvan and Newman. Computing the information centrality of an edge requires the calculation of the distances between all pairs of vertices, which can be done with breadth-first-search in a time O(mn). So, in order to compute the information centrality of all edges one requires a time O(m2n). At this point one removes the edge with the largest value of information centrality and recalculates the information centrality of all remaining edges with respect to the running graph. Since the procedure is iterated until there are no more edges in the network, the final complexity is O(m3n), or O(n4) on a sparse graph. The partition with the largest value of modularity is chosen as most representative of the community structure of the graph. The method is much slower than the algorithm of Girvan and Newman. Partitions obtained with both techniques are rather consistent, mainly because information centrality has a strong correlation with edge betweenness. The algorithm by Fortunato et al. gives better results when communities are mixed, i. e. with a high degree of interconnectedness, but it tends to isolate leaf vertices and small loosely bound subgraphs.

  4. To extend to PT: kListEh PT2hk=Ek after zeroing the h bit of Ek PTG1 E3 pred=(NPZ)|(PZ&AcyclicPathEnd) APTG1 APTG1 E3predicate = (NPZ&NotCycleEnd)| (PZ&AcyclicPathEnd) PTG1, extension of EG1 E 2-lev stri=|V|=4 SPTG1 Edge pTree, E, PathTree(PT), ShortestPathvTree(SPT),AcyclicPathTree(APT) andCycleList(CL) of G1 G1 E3key 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 PE3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 PE2 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 E2key 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 1111 kListPT2hj PT3hjk=Ek after zeroing Ek j bit. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2 1 APTG1 1 1 1 1 PTG1 1 1 1 1 (pred is NotPureZero) First, construct stride=|V|, 2-level Edge pTree, all others are constructed concurrently from it. 2LEG1 1 1 1 1 1 0 0 1 1 2 0 0 0 1 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 SPTG1, init E1=SP1,1E2=SP2,1E3=SP3,1 E4=SP4,1 4 1 1 1 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 kListPT3hij PT4hijk=Ek after zeroing i and j bits of Ek E 2levstr=4 pred=NPZ All are 3 hop cycles. Each has 3 start pts , 2 directions. Each repeat 6 times. 6/6=1 3hop cycles (1341) 1 1 0 0 1 1 2 1 0 0 0 1 3 1 1 0 0 1 4 1 1 1 1 0 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 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 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 3 4 CLG1 1 0 2 1 1 2 2 0 2 1 3 1 2 0 1 E one-level E 1lev, pred=NPZ 1341 3413 1431 PE1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 EG1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 E1 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 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 4134 3143 4314 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 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 1 0 0 2 2 1 0 1 0 3 2 0 1 0 0 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 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 SPT is completed. For Big Graphs, could stop here (e.g., Friends has ~1B vertices but a diameter of 4, so we would only need to build PT 4-hop paths) and possible expressed as a tree of lists rather than a tree of bitmaps. For sparse BigGraphs, E could be leveled further and/or a tree of lists (then APT, SPT will be also). SPT(G)k (with k turned on) is mask (>0 is “yes”) for connectivity comp, COMP(G)kvk. For bitmap of COMPkbitslicing SPT (SPTk,h..SPTk,0k=1..|V| then COMPk ORj=h..0SPTk,h. SPT structure may be useful as separate “categorical” bitmaps  Shortest Path Length (SPk,h h=1..H. Also keep a mask of Shortest Paths so far, SPSFk vertex, k. With each new SP bitmap, SPB, SPSFkSPSFk| SPB, SPk,h+1 SPB & SPSFk. 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , SPSFk 1 0 0 1 1 2 0 0 0 1 3 1 0 0 1 4 1 1 1 0 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 4 0 1 1 0 3 1 0 0 0 1 4 1 0 0 1 0 4 3 1 0 0 0 1 3 0 0 0 1 2 4 1 0 1 0 3 4 1 1 0 0 1 3 4 1 1 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 1 4 3 1 0 0 0 3 1 4 0 1 1 0 4 3 1 0 0 0 1 SPTgives the Connectivity Component Partition; Maximal Cliques (go across SPk,1 then look within subsets of those k’s for commonality); Note, Cliques are 0-plexes. Each mask, SPk,1 masks a 1-plex. Each SPk,1&SPk,2 masks a 2-plex (which is SPSFk,2? So if we save each SPSF instead of overwriting, we have k-plex masks w/o further work?), etc. Next construct predicates for each Path related data structures, PT APT SPT SPSF, to make them into pTrees on a k-path table, E, E2, E3, … 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 1,1,1 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 E PT SPTAPT of graph as predicate Trees on E(MaxPathLength). 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , SP3,1 SP4,1 SPVertex=3, Len=2 SP3,1|2 SP4,1|2 SP1,1 SP2,1 SP1,2 SP2,2 SP1,1|2 SP2,1|2 2 1 0 0 0 1 3 1 1 0 0 1 4 1 1 1 1 0 1 1 0 0 1 1 2 2 1 0 1 0 3 2 0 1 0 0 2 12 1 0 1 1 3 12 1 1 0 1 4 12 1 1 1 0 1 2 0 1 0 0 1 12 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

  5. 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 6 G3 6 G4 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 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

  6. TheEdgepTree(E), PathTree(PT), ShortestPathvTree(SPT),AcyclicPathTree(APT) andCycleList(CL) of a graph, G5 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 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 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 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 2 3 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 7 3683 5 6 3863 5175 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 6386 6836 7157 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 8638 PT Clique Miner Algorithm A clique is all cycles Extend to a k-plex(k-core) mining algorithm? PT(=APT+CL), SPT are powerful datamining tools with closure properties (to eliminate branches) . SPTG5 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 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 are 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 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. k-plex (missing  k edges) mining alg? k-core (has  k edges) mining alg? Density (internal edge density >> external|avg) mining alg? Degree (internal vertex degree >> external|avg) mining alg? DiameterG5 is max{Diameterk} = 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.

  7. E=A1Ps 8 0 0 0 1 0 0 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 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 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 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 c g A6Ps A5Ps A4Ps A2Ps cycles in blue (not in APT) 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 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 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 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 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 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 1 1 1 1 1 1 1 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 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 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 2 1 0 1 1 0 1 0 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 0 0 1 1 0 0 0 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 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 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 4 1 1 1 0 0 1 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 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 5 1 0 1 0 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 1 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 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 0 1 0 1 0 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 1 1 1 1 1 0 1 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 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 1 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 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 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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 2 1 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

  8. 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 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 All Shortest Path pTrees for a unipartite undirected graph, G7 (SP1, SP2, SP3, SP4, SP5) 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 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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 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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

  9. Trying Hamming Similarity to detect communities on G7 and G8 1 5 4 2 3 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 Zachary's karate club, a standard benchmark in community detection. (best partition found by optimizing modularity of Newman and Girvan) 41 46 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 42 8 45 47 7 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 =2deg 44 6 43 40 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 =3deg 9 39 8 8 8 8 8 8 9 10 8 8 8 8 8 8 8 10 8 =4deg 38 53 48 12 52 1 1 8 1 1 1 1 1 1 =5deg G8 10 13 To produce an [all?] actual shortest path[s] between x and y: Thm: To produce a [all?]: S2P[s], take a [all?] middle vertex[es], x1, from SP1x & SP1y, produce: xx1y; S3P[s], take a [all?] vertex[es], x1, from SP1x and a [all?] vertex[es], x2, from S2P(x1,y): xx1x2y etc. Is it productive to actually produce (one time) a tree of [all?] shortest paths? I think it is not! Hamming similarity: S(S1,S2)=DegkDif(S1,S2) 14 11 17 1 2 3 4 1 2 3 4 5 6 7 36 54 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 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 24 35 15 23 22 14 20 20 17 15 16 24 30 27 18 39 28 42 Can see that this Works Poorly At 1. 37 21 49 17 17 25 17 2 2 17 24 18 1 14 3 24 7 7 25 Not working! On the other hand, our standard community mining techniques (for kplexes) worked well on G7. Next slide let’s try Hamming on G8. 19 34 20 27 25 18 50 51 26 30 29 28 33 31 32 G7 3 7 12 3 25 Deg1 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 19 Deg2 5 8 12 17 8 16 17 16 4 24 21 21 26 20 20 20 19 16 19 23 30 13 15 22 14 20 18 11 14 15 10 15 14 21 14 17 10 4 3 2 4 3 10 21 8 10 15 18 15 15 10 17 18 35

  10. G9, Agglomerative clustering of ESP2 using Hamming Similarity In ESP2, using Hamming similarity, we get three Event clusters, clustering events iffpTrees [Hamming] identical: EventCluster1={1,2,3,4,5} EventCluster2={6,7,8,9} EventCluster3={10,11,12,13,14} ESP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 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 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 The Degree % of affiliation of Women with R,G,B events is: RGB 1 100% 75% 0% 2 80% 75% 0% 3 80% 100% 0% 4 80% 75% 0% 5 60% 25% 0% 6 40% 50% 0% 7 20% 75% 0% 8 0% 75% 0% 9 20% 75% 0% 10 0% 75% 20% 11 0% 50% 40% 12 0% 50% 80% 13 0% 75% 80% 14 0% 75% 100% 15 0% 50% 60% 16 0% 50% 0% 17 0% 25% 20% 18 0% 25% 20% W WSP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W WSP3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 6 7 6 6 10 10 10 11 10 10 10 8 7 6 9 12 12 12 8 7 8 8 4 4 4 3 4 4 4 6 7 8 5 2 2 2 3 3 6 4 8 8 10 14 12 5 4 6 3 3 WSP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ESP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 1 1 1 1 1 1 1 0 0 0 0 0 ESP3=ESP1’ and ESP4=ESP2’ so again, in this case, all info is already available in ESP1 and ESP2 (all shortest paths are of length 1 or 2). We don’t need ESPk k>2) G9 9 9 9 9 9 e e e e 9 9 9 9 9 18 16 18 18 12 16 16 17 18 18 17 17 18 18 18 17 13 13 Clustering Women using Degree% RGB affiliation: WomenClusterR={1,2,4,5} WomanClusterG={3,6,7,8,9,10,11,16,17,18} WomanClsuterB={12,13,14,15} WSP3=WSP1’ and WSP4=WSP2’ so, in this case, all information is already available in WSP1 and WSP2 (All shortest paths are of length 1 or 2) (We don’t need WSPk k>2) This clustering seems fairly close to the authors. Other methods are possible and if another method puts event6 with 12345, then everything changes and the result seem even closer to the author’s intent..

  11. K-plex search on G9 (A k-plex is a SG missing  k edges If H is a k-plex and F is a ISG, then F is a kplex A graph (V,E) is a k-plex iff |V|(|V|-1)/2 – |E| k WSP2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Events123456789abcde 14*13/2=91 degs=88888dddd88888 |Edge|=66 kplex k25 h f h f b f f g h h g g h h h g c c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Events23456789abcde Not calculating k degs=7777cccc88888 Until it gets lower ESP2 1 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 0 1 1 1 1 1 1 1 1 0 0 0 0 0 Events3456789abcde14*13/2=91 degs=666bbbb88888 |Edges|=66 kpl Events456789abcde14*13/2=91 degs=55aaaa88888 |Edges|=66 kplex k25 Women123456789abcdefghi 18*17/2=153 degs=hfhfbffghhgghhhgcc |Edges| =139 kplex k14 Events56789abcde14*13/2=91 degs=4999988888 |Edges|=66 kplex k25 Women123456789abcdefgh18*17/2=153 degs=gfgfbfffggffgggfc |Edges| =139 kplex k14 Events6789abcde 9*8/2=36 A 9Clique! degs=888888888 |Edges|=36 kplex k0 Women123456789abcdefg18*17/2=153 degs=ffffbffeffeefffe |Edges| =139 kplex k14 8 8 8 8 8 d d d d 8 8 8 8 8 So take out {6789abcde} and start over. 17 15 17 15 11 15 15 16 17 17 16 16 17 17 17 16 12 12 Women12346789abcdefg 15*14/2=105 degs=eeeeeeeeeeeeeee |Edges| =105 15kplex k0 15Clique Events12345 5*4/2=10 |Edges|=10 kplex k 0 A 5clique! degs: 44444 So take out {12346789abcdefg} and start over. If we had used the full algorithm which pursues each minimum degree tie path, one of them would start by eliminating 14 instead of 1. That will result in the 9Clique 123456789 and the 5Clique abcde. All the other 8 ties would result in one of these two situations. How can we know that ahead of time and avoid all those unproductive minimum degree tie paths? Women5hi 3*2/2=3 degs=011 |Edges| =1 kplex k2 G9 Womenhi 2*1/2=1 degs=11 |Edges| =1 kplex k0 Clique We get no information from applying our kplex search algorithm to WSP2. Again, how could we know this ahead of time to avoid all the work? Possibly by noticing the very high 1-density of the pTrees? (only 28 zeros)? Every ISG of a Clique is a Clique so 6789 and 789 are Cliques (which seems to be the authors intent?) If the goal is to find all maximal Cliques, how do we know that CA=123456789 is maximal? If it weren’t then there would be at least one of abcde which when added to CA=123456789 would results in a 10Clique. Checking a: PCA&Pa would have to have count=9 (It doesn’t! It has count=5) and PCA(a) would have to be 1 (It isn’t. It’s 0). The same is true for bcde. The same type of analysis shows 6789abcde is maximal. I think one can prove that any Clique obtained by our algorithm would be maximal (without the above expensive check), since we start with the whole vertex set and throw out one at a time until we get a clique, so it has to be maximal? The Women associated strongly with the blue EventClique, abgde are {12 13 14 15 16} and associated but loosely are {10 11 17 18}. The Women associated strongly with the green EventClique, 12345 are {1 2 3 4 5} and associated but loosely are {6 7 9}

  12. G10 E=SP1 2level pTrees LevelOneStride=19 (labelled 0-i), Level0Stride=10 (labelled 0-9) Note: SP1 should be called S1PDV for “Shortest 1 Path Destination Verticies, because each one, e.g. S1PDV(v1) maps all such destination verticies from that given starting vertex, v1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 21 1 1 1 1 OutDeg 1 8 1 0 0 0 0 1 9 1 0 0 0 0 2 0 1 0 0 0 0 2 1 1 0 0 0 0 2 2 1 0 0 0 0 2 3 1 0 0 0 0 2 4 1 0 0 0 0 2 5 1 0 0 0 0 2 6 1 0 0 0 0 2 7 0 1 0 0 0 2 8 0 1 0 0 0 2 9 0 1 0 0 0 3 0 0 1 0 0 0 3 1 0 1 0 0 0 3 2 0 1 0 0 0 3 3 0 1 0 0 0 3 4 0 1 0 0 0 3 5 0 1 0 0 0 3 6 0 1 0 0 0 3 7 0 0 1 0 0 3 8 0 0 1 1 1 3 9 0 0 1 0 0 4 0 0 0 1 0 0 4 2 0 0 0 0 1 4 3 0 0 0 0 0 0 0 1 0 0 0 1 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 0 0 1 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 0 0 1 0 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 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 0 0 1 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 0 0 1 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 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1 1 1 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 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 b7 1 1 1 1 1 1 1 0 0 0 a3 5 3 4 8 5 4 4 8 5 7 6 9 5 8 7 0 5 9 6 8 6 0 6 7 6 1 6 6 6 3 6 6 6 5 e 7 6 6 6 1 7 1 4 9 b8 tens dig 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 4 5 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 4 7 0 0 0 0 1 0 0 1 0 0 5 1 0 0 0 0 1 0 0 0 0 0 5 2 0 0 0 0 1 1 1 0 0 1 4 6 0 0 0 0 1 1 0 1 0 0 4 8 0 0 0 0 1 4 9 0 0 0 0 1 0 0 1 0 0 5 5 0 0 0 0 1 5 6 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 7 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 3 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 b6 b5 b9 a5 c3 a4 1 1 1 1 0 1 1 0 0 1 c2 c4 2 1 3 4 a7 5 G10 a6 a8 a9 b4 6 b0 c1 c0 7 19 20 8 18 22 97 98 99 21 23 a1 90 24 9 35 a0 84 a2 25 b1 units 0 1 2 3 4 5 6 7 8 9 85 36 40 89 10 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 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 1 0 0 0 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 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 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 37 26 92 76 83 27 86 b3 28 93 38 91 11 82 75 70 94 42 29 87 41 30 12 69 80 88 39 b2 57 81 43 31 32 58 77 95 13 56 h9 34 units 0 1 2 3 4 59 68 c5 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 44 79 0 1 0 0 0 1 1 0 0 0 33 96 14 78 60 55 15 52 51 67 16 45 61 50 h8 units 0 1 2 3 4 5 6 7 8 9 17 46 i0 62 63 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 54 0 1 0 0 1 49 66 h7 65 53 64 48 h3 47 h5 h6 h4 h2 c7 c6 h0 71 h1 74 g9 72 units 0 1 2 3 4 g2 73 g3 g6 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 g8 c8 d4 g4 d3 g5 c9 g7 d2 g1 e6 d1 e7 d0 units 0 1 2 3 4 e5 g0 e2 0 1 0 0 0 e4 d7 d5 f9 f7 e3 f8 e1 f6 d6 e8 e0 d8 d9 units 0 1 2 3 4 5 6 7 8 9 e9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 G10: Web graph of pages of a website and hyperlinks. Communities by color (Girvan Newman Algorithm). |V|=180 (1-i0) and |E|=266. Vertices with OutDeg=0 (leaves) do not have pTrees shown because pTrees display only OutEdges and thus those OD=1 have a pure0 pTree. 0 0 0 0 0 0 0 1 0 0 f0 f1 f5 f2 f4 f3

  13. 7OD G10 E=SP1 2level pTrees LevelOneStride=19 (labelled 0-i), Level0Stride=10 (labelled 0-9) 9 OD L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i C 4 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i H 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 211 1 1 1 OutDeg 1 1 2 OD B 5 C 4 B 6 B 7 1 8 1 0 0 0 0 1 9 1 0 0 0 0 2 0 1 0 0 0 0 2 1 1 0 0 0 0 2 2 1 0 0 0 0 2 3 1 0 0 0 0 2 4 1 0 0 0 0 2 5 1 0 0 0 0 2 6 1 0 0 0 0 2 7 0 1 0 0 0 2 8 0 1 0 0 0 2 9 0 1 0 0 0 3 0 0 1 0 0 0 3 1 0 1 0 0 0 3 2 0 1 0 0 0 3 3 0 1 0 0 0 3 4 0 1 0 0 0 3 5 0 1 0 0 0 3 6 0 1 0 0 0 3 7 0 0 1 0 0 3 8 0 0 1 1 1 3 9 0 0 1 0 0 4 0 0 0 1 0 0 4 2 0 0 0 0 1 4 3 0 0 0 0 0 0 0 1 0 0 B 4 C 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 7 5 7 7 7 6 7 7 7 7 7 6 h 5 L1 0 1 2 3 4 5 6 7 8 9 a b . . . L0 0 1 2 3 4 5 6 7 8 9 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 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 0 0 1 0 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 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 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 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1 1 1 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 1 0 0 0 0 C 0 B 2 C 4 C 1 7 7 C 4 C 2 B 9 C 4 C 3 B 8 5 5 2 8 20 4 OD L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i 7 8 0 0 0 0 1 0 0 0 1 0 7 9 0 0 0 0 1 0 0 0 1 1 8 0 0 0 0 1 0 0 0 0 1 0 8 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 9 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 9 1 4 0 4 6 7 6 H 8 H 9 B 1 B 4 C 6 C 7 A 7 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 8 6 8 0 8 7 7 9 8 9 8 5 9 0 A 6 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 1 1 1 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 OD L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 5 1 4 6 5 3 4 8 5 4 4 8 5 5 4 9 5 7 6 9 5 8 7 0 5 9 6 8 6 0 6 7 6 1 6 6 6 3 6 6 6 5 e 7 6 6 6 1 7 1 4 9 9 2 9 1 9 3 9 1 9 5 7 9 9 6 7 8 6 6 2 3 3 3 17 3 2 2 2 2 OD L0 0 1 2 3 4 5 6 7 8 9 4 6 0 0 0 0 1 1 0 1 0 0 4 8 0 0 0 0 1 4 9 0 0 0 0 1 0 0 1 0 0 L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i 4 5 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 4 7 0 0 0 0 1 0 0 1 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 5 2 0 0 0 0 1 1 1 0 0 1 5 6 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 7 2 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 3 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 7 4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 C 5 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 9 7 A 7 9 8 9 1 9 9 8 8 A 0 A 8 1 1 1 1 0 1 1 0 0 1 L0 0 1 2 3 4 5 6 7 8 9 0 1 0 1 0 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 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 A 1 A 9 A 2 B 0 A 4 A 7 L0 0 1 2 3 4 5 6 7 8 9 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 8 8 7 8 9 1 9 4 9 1 5 2 A 5 A 3 A 7 20 OD L0 0 1 2 3 4 5 6 7 8 9 L1 0 1 2 3 4 5 6 7 8 9 a b c d e f g h i D 2 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 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 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 1 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 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 0 0 1 0 0 0 H 4 D 2 4 6 7 2 7 3 7 4 C 6 C 7 C 8 C 9 C 6 9 1 D 2 C 7 9 1 D 2 C 8 7 9 D 2 C 9 7 8 D 0 H 4 D 2 D 1 4 5 7 8 D 2 D 4 D 2 D 3 D 2 4 6 8 1 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 L0 0 1 2 3 4 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 D 5 D 2 D 6 F 5 D 2 D 7 D 9 D 2 D 8 F 4 D 2 D 9 9 1 D 2 E 0 9 1 D 2 E 1 7 9 D 2 E 2 D 2 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 L0 0 1 2 3 4 5 6 7 8 9 0 1 0 1 0 1 0 0 1 1 1 2 1 OD E 3 F 0 D 2 E 4 E 9 D 2 E 5 E 8 D 2 E 6 E 7 D 2 L0 0 1 2 3 4 B 1 B 2 B 2 7 6 h 1 B 3 B 2 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 0 0 0 L0 0 1 2 3 4 0 1 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 H 6 F 7 G 1 G 2 G 4 G 5 G 3 I 0 H 5 H 3 G 6 H 9 H 2 F 6 H 0 H 1 H 8 G 7 G10 leaves (OutDegree=0): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 41 44 62 64 67 68 69 70 82 83 84 85 a3 a6 a8 a9 b0 B7 b8 b9 e7 e8 e9 f0 f1 f2 f3 f4 f5 f8 f9 g0 g8 g9 h7 G 6 H 1 F 8 F 7 7 7 H 4 G 0 F 6 G 1 H 4 H 0 H 4 I 0 G 1 H 4 G 7 9 1 H 4 4 4 H 4 H 9 I 0 4 3 7 8 7 9 F 9 F 7 H 4 H 4

  14. C640 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 E040 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 D9 E1 E2 E3 E4 E5 E6 H8 H9 D246 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 18 1 G10 SP1 Lists 7577 SP2 Lists 8021 22 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41 42 43 45 46 49 78 79 83 84 85 7677 19 2 36 2 7776 H5 37 3 6 20 3 38 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 47 48 50 51 71 74 78 B2 D1 H7 7841 43 46 81 88 21 4 D3D2 22 5 7946 81 87 95 96 D4D2 23 6 8141 43 46 81 87 88 95 96 8038 81 24 7 D5D2 E146 72 73 74 81 87 95 96 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E2 E3 E4 E5 E6 C740 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 8178 79 83 84 85 25 8 8638 81 D6E5 D2 39 12 26 9 8680 8746 81 95 96 40 10 27 10 D7D9 D2 8840 41 43 46 76 81 86 88 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 8779 42 46 51 78 B2 D1 H7 28 11 D8E4 D2 8878 91 29 12 E246 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E3 E4 E5 E6 D991 D2 30 13 43 41 46 81 88 8985 31 14 45 41 43 45 47 48 50 76 D2 H1 81 88 E091 D2 90A6 C846 72 73 74 81 87 95 96 C6 C7 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 32 15 9127 43 45 47 48 50 51 77 78 79 80 83 84 85 88 91 A6 A7 A8 A9 B0 B2 C4 D2 H4 I0 9140 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 E179 D2 E346 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E4 E5 E6 33 16 34 17 E2D2 46 46 49 55 72 78 B2 D1 H6 H7 35 18 E3F0 D2 36 19 9291 C941 43 46 81 88 E446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E5 E6 E4E9 D2 37 20 23 47 46 49 D2 D046 72 73 74 81 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H0 H1 H2 H3 H4 H5 H6 H7 H8 9240 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 38 21 22 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41 42 43 45 46 49 9391 48 45 48 50 51 72 74 E5E8 D2 9491 52 E6E7 D2 49 45 47 48 49 50 51 D2 D546 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E6 9579 H3 G1 H8 H6 H5 H2 F6 H1 I0 H0 G6 H3 G5 H4 H4 G5 I0 F7 G1 H9 G2 G3 H8 G4 H6 G6 H9 G7 H0 H5 H1 H2 43 46 78 79 81 H0 H1 H2 H3 H5 H7 H8 40 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H0 H1 H2 H3 H5 H6 H7 H9 F8 G1 H4 G1 H4 F6 F7 G7 G0 F9 H1 G6 I0 H0 H4 41 44 46 81 87 88 95 96 H4 F7 H4 46 81 G1 H0 H1 H3 H4 H5 H6 H7 H8 46 81 H0 H2 H3 H4 H5 H6 H7 H8 44 H4 H9 H4 46 81 H0 H1 H2 H3 H5 H7 H8 77 H4 46 76 81 H0 H1 H2 H3 H6 H7 H8 G6 45 47 48 50 51 77 91 G1 I0 46 81 H0 H1 H2 H3 H5 H6 H7 H8 H4 46 81 H0 H1 H2 H3 H4 H5 H6 H7 H8 46 81 H1 H2 H3 H4 H5 H6 H7 H8 44 H4 H9 91 H4 I0 43 78 79 9340 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 9678 50 45 47 48 49 50 H4 39 29 D141 43 46 72 73 74 81 88 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 97A7 40 27 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 42 45 9891 51 45 47 48 50 9440 46 53 54 55 56 57 58 59 60 61 62 63 64 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 43 78 9988 52 45 47 48 49 50 51 66 67 68 69 70 75 B3 45 46 51 78 B2 D1 H7 D245 47 48 49 78 79 91 E7 E8 E9 F0 A0A8 A1A9 46 45 47 48 50 51 53 46 47 49 D346 72 73 74 C6 C7 C8 C9 D1 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 A2B0 47 48 72 54 46 47 49 A4A7 48 46 47 49 9546 81 87 96 55 46 71 74 A5A3 A7 49 46 71 74 9641 43 46 81 88 56 46 51 55 77 B2 H6 D446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 A798 99 A4 A5 9798 99 A4 A5 50 46 51 55 H6 B1 B2 9840 46 76 81 86 89 90 97 99M A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 63 1 51 46 B276 H1 52 46 53 54 55 56 57 58 59 60 61 62 63 64 49 46 74 D546 72 73 74 C6 C7 C8 C9 D1 D3 D8 D9 E0 E1 E2 E3 E4 E5 E6 B3B2 72 46 48 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B4C4 9978 91 53 48 A498 99 A5 B5C4 54 48 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 E8 A598 99 A4 B6B7 55 49 A788 91 A3 73 46 47 49 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B1 76 H1 C0B2 C4 56 50 75 B3 B277 G1 H4 C177 C4 57 69 B3 76 H1 D746 72 73 74 91 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B4B5 B6 C0 C1 C2 C3 C2B9 C4 58 70 C3B8 74 46 49 71 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 59 68 B5B4 B6 C0 C1 C2 C3 C4B4 B5 B6 C0 C1 C2 C3 60 67 C076 B4 B5 B6 C1 C2 C3 H1 61 66 D846 72 73 74 C6 C7 C8 C9 D1 D3 D5 D9 E0 E1 E2 E3 E4 E5 E6 E9 c5 45 D5 OD=0: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 41 44 62 64 67 68 69 70 82 83 84 85 a3 a6 a8 a9 B0 B7 b8 b9 e7 e8 e9 f0 f1 f2 f3 f4 f5 f8 f9 g0 g8 g9 h7 63 66 C177 B4 B5 B6 C0 C1 C2 C3 H5 7576 H5 C691 D2 65 E7 7676 H5 C2B4 B5 B6 C0 C1 C3 D940 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 E0 E1 E2 E3 E4 E5 E6 H8 H9 77H4 C791 D2 66 61 7845 47 48 50 51 78 79 83 85 91 71 49 C879 D2 C477 B2 B7 B8 B9 C4 C978 72 47 D2 7946 47 48 50 51 78 79 83 84 85 C5 46 51 78 B2 D1 D2 H7 73 48 D2 D0H4 D2 74 49 D2 D178 D2

  15. Horizontal Trans tbl TT(I) t1 i1 t2 i1, i2, i4 t3 i1, i3 t4 i1, i2, i4 t5 i3, i4 Its graph t1 i1 t2 i2 A t3 i3 t4 i4 C t5 T I AssociationRule Mining (ARM) The support [ratio] of itemsetA,supp(A), is the fraction of Ts such that A  T(I), e.g., if A={i1,i2} and C={i4} then supp(A) = |{t2,t4}| / |{t1,t2,t3,t4,t5}| = 2/5 Note: | | means set size. The support [ratio] of ruleAC, supp(AC),is the support of {A C}=|{T2,T4}|/|{T1,T2,T3,T4,T5}|=2/5 The confidence of ruleAC, conf(AC),is supp(AC) / supp(A) = (2/5) / (2/5) = 1 Data Miners typically want to find all STRONG RULES, AC, with supp(AC) ≥ minsupp and conf(AC) ≥ minconf (minsupp, minconf are threshold levels) Note that conf(AC) is also just the conditional probability of t being related to C, given that t is related to A). Given a two entity relationship, we can do ARM with either entity taking the role of the transaction set Given any relationship between entities, T (e.g., a set of Customer Transactionss which are involved in those relationship instances). and I (e.g., a set of Items which are involved in those relationship instances). The itemset, T(I),associated with (or related to) a particular transaction, T, is the subset of the items found in the shopping cart or market basket that the customer is bringing through check out at that time). An Association Rule, AC, associates 2 disjoint Itemsets. (A=antecedent, C=consequent) APRIORI Association Rule Mining:Given a Transaction-Item Relationship, the APRIORI algorithm for finding all Strong I-rules can be done by:Processing a Horizontal Transaction Table (HTT) through vertical scans to find all FrequentI-sets ( e.g., I-sets "frequently" found in baskets).Processing a Vertical Transaction Table (VTT) through horizontal operations to find all FrequentI-setsThen each Frequent I-set found is analyzed to determine if it is the support set of a strong rule.Finding all Frequent I-sets is the hard part. To do this efficiently, the APRIORI Algorithm takes advantage of the "downward closure" property for Frequent I-sets: If a I-set is frequent, then all its subsets are also frequent.E.g., in the Market Basket Example, If A is an I-subset of B and if all of B is in a given Transaction's basket, the certainly all of A is in that basket too. Therefore Supp(A)  Supp(B) whenever AB.First, APRIORI scans to determine all Frequent 1-item I-sets (contain 1 item; therfore called 1-Itemsets),next APRIORI uses downward closure to efficiently find candidates for Frequent 2-Itemsets,next APRIORI scans to determine which of those candidate 2-Itemsets is actually Frequent,next APRIORI uses downward closure to efficiently find candidates for Frequent 3-Itemsets,next APRIORI scans to determine which of those candidate 3-Itemsets is actually Frequent, ...Until there are no candidates remaining (on the next slide we walk through an example using both a HTT and a VTT)

  16. F3 = L3 C1 F1 = L1 C2 F2 = L2 C3 C2 itemset Scan D {2 3 5} {1 2 3} {1,3,5} Scan D Scan D P1 2 //\\ 1010 P2 3 //\\ 0111 P1^P2^P3 1 //\\ 0010 P1^P2 1 //\\ 0010 Build Item Ptrees: Scan D P3 3 //\\ 1110 P1^P3 ^P5 1 //\\ 0010 P1^P3 2 //\\ 1010 P4 1 //\\ 1000 P2^P3 ^P5 2 //\\ 0110 P5 3 //\\ 0111 P1^P5 1 //\\ 0010 F2={13}{23}{25}{35} cts 2 2 3 2 F1={1}{2}{3}{5} cts: 2 3 3 3 F3={235} cts 2 P2^P3 2 //\\ 0110 P2^P5 3 //\\ 0111 P3^P5 2 //\\ 0110 HTT {123} pruned since {12} not frequent {135} pruned since {15} not frequent It seems the pruning step in purple above is unnecessary here since root count will show up below the threshold and that root count (using PopCount) is almost free anyway??? Example ARM using uncompressed ItemPtrees (the 1-count at the root of each Ptree) All we need to do ARM are theses FrequentItemTables with Counts.

  17. L3 L1 L2 Data_Lecture_4.1_ARM 1-ItemSets don’t support Association Rules (They will have no antecedent or no consequent). 2-Itemsets do support ARs. Are there any Strong Rules supported by Frequent=Large 2-ItemSets(at minconf=.75)? {1,3} conf({1}{3}) = supp{1,3}/supp{1} = 2/2 = 1 ≥ .75 STRONG conf({3}{1}) = supp{1,3}/supp{3} = 2/3 = .67 < .75 {2,3} conf({2}{3}) = supp{2,3}/supp{2} = 2/3 = .67 < .75 conf({3}{2}) = supp{2,3}/supp{3} = 2/3 = .67 < .75 {2,5} conf({2}{5}) = supp{2,5}/supp{2} = 3/3 = 1 ≥ .75STRONG! conf({5}{2}) = supp{2,5}/supp{5} = 3/3 = 1 ≥ .75STRONG! {3,5} conf({3}{5}) = supp{3,5}/supp{3} = 2/3 = .67 < .75 conf({5}{3}) = supp{3,5}/supp{5} = 2/3 = .67 < .75 Are there any Strong Rules supported by Frequent or Large 3-ItemSets? {2,3,5} conf({2,3}{5}) = supp{2,3,5}/supp{2,3} = 2/2 = 1 ≥ .75STRONG! conf({2,5}{3}) = supp{2,3,5}/supp{2,5} = 2/3 = .67 < .75 No subset antecedent can yield a strong rule either (i.e., no need to check conf({2}{3,5}) or conf({5}{2,3}) since both denominators will be at least as large and therefore, both confidences will be at least as low. conf({3,5}{2}) = supp{2,3,5}/supp{3,5} = 2/3 = .67 < .75 No need to check conf({3}{2,5}) or conf({5}{2,3}) DONE!

  18. ESP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E Using ARM to find kplexes on the bipartite graph, G9? Does it work? 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 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 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0 1 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 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 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 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 WSP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18W 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 WSP2 3 3 6 4 8 8 a e c 5 4 6 3 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ESP2 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 1 1 1 1 1 1 1 0 0 0 0 0 G9 9 9 9 9 9 e e e e 9 9 9 9 9 18 16 18 16 12 16 16 17 18 18 17 17 18 18 18 17 13 13 WomenSet ARM: MinSup=6 Mincnf=.75 EventSet ARM MnSp=9 Mncf=.75 Frequent 1WomenSets: 1 2 3 4 12 13 14 Frequency (#events attended) 8 7 8 8 6 7 8 Freq 1EventSets: 3 5 6 7 8 9 c Freq (# attended) 6 8 8 10 14 12 6 Candidate 2WomenSets: 12 13 14 1c 1d 1e 23 24 2c 2d 2e 34 3c 3d 3e 4c 4d 4e cd ce de Freq #events co-attended 6 7 7 2 2 2 6 6 1 2 2 7 2 3 3 2 3 3 6 5 6 Cand 2EventSets: 35 36 37 38 39 3c 56 57 58 59 5c 67 68 69 6c 78 79 7c 89 8c 9c Freq=#attended 6 5 4 5 2 0 6 6 7 3 0 5 7 4 1 8 5 4 9 5 5 Frequent 2WomenSets: 12 13 14 1c1d 1e 23 24 2c 2d 2e 34 3c 3d 3e 4c 4d 4e cd ce de Freq #events co-attended 6 7 7 22 2 6 6 1 2 2 7 2 3 3 2 3 3 6 5 6 freq 2EventSets: 35 36 37 38 39 3c 56 57 58 59 5c 67 68 69 6c 78 79 7c 89 Freq=#attended 6 5 4 5 2 0 6 6 7 3 0 5 7 4 1 8 5 4 9 Cand 3EventSets 568 578 all others excl because a sub2 not freq Freq # attended 6 5 Cand3WSets: 123 124 134 234 (cde is excluded since ce is infreq) Freq #events co-attended 5 5 6 5 Strong Erules 35 53 56 65 57 58 68 78 98 567 657 567 576 675 (Says 567 is a strong Event community?) Frequent 3WomenSets:123 124 134 234 Freq #events co-attended 5 5 6 5 Freq 3ESets: 567 Freq=6 5 StrongWrules21 12 13 31 14 41 23 32 24 42 34 43 134 314 413 134 143 341 Says 1234 is a strong Women community? Confidence: .83 .75 .87 .87 .87 .87 .83 .75 .83 .75 .87 .87 .75 .75 .75 .83 .83 .83 But 134 is a very strong Women Commun? Note: When I did this ARM analysis, I had several degrees miscounted. None-the-less, I think the same general negative result is expected. Next we try using the WSP2 and ESP2 relationships for ARM??

  19. WSP2 ESP2 EventSet ARM MnSp=9 Mncf=.75 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 WSP2 WomenSet ARM MinSup=18 Mincnf=.75 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 W 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ESP2 Freq1WSets: 1349adef Frequencies all 18 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 E 1 1 1 1 1 1 1 1 1 0 0 0 0 0 C2WSets: 13 14 19 1a 1d 1e 1f 34 39 3a 3d 3e 3f 49 4a 4d 4e 4f 9a 9d 9e 9f ad ae af de dfef Freq all 18 This is not interesting! Go to ESP2 Eset ARM G9 9 9 9 9 9 e e e e 9 9 9 9 9 18 16 18 16 12 16 16 17 18 18 17 17 18 18 18 17 13 13 Freq1EventSets E1 Freq Freq2EventSets E1 E2 Freq c de 99 b cde 999 a bcde 9999 d e 9 9 abcde 99999 7 89abcde ee99999 8 9abcde e99999 123456789abcde 99999eeee99999 1 23456789 99999999 6 789abcde eee99999 2 3456789 9999999 3 456789 999999 4 56789 99999 5 6789 9999 Freq3EventSets E1 E2 E3 Freq 1 2 3456789 9999999 6|7|8|9|a b cde 999 1|2 3 456789 999999 1|2|3 4 56789 99999 1|2|3|4 5 6789 9999 1|2|3|4|5 6 789 999 1|2|3|4|5 7 89 99 6 7 89abcde ee99999 6|7 8 9abcde e99999 6|7|8|9 a bcde 9999 1|2|3|4|5 8 9 9 6|7|8 9 abcde 99999 6|7|8|9|a|b c de 99 6|7|8|9|a|b|c d e 9 All rule confidences are either 100% (9/9 or e/e) or 9/e=64% Freq6EventSets E1 E2 E3 E4 E5 E6 Freq Freq7EventSets E1 E2 E3 E4 E5 E6 E7 Freq Freq8EventSets E1 E2 E3 E4 E5 E6 E7 E8 Freq Freq9EventSets E1 E2 E3 E4 E5 E6 E7 E8 E9 Freq Freq5EventSets E1 E2 E3 E4 E5 Freq Freq4EventSets E1 E2 E3 E4 Freq 6|7|8 9 a b c d e 9 6|7|8 9 a b c de 99 6|7|8 9 a b cde 999 6|7|8 9 a bcde 9999 6|7|8|9 a b c d e 9 6|7|8|9 a b c de 99 6|7|8|9 a b cde 999 1 2 3 4 56789 99999 1 2 3 4 5 6789 9999 1 2 3 4 5 6 789 999 1 2 3 456789 999999 1 2 3 4 5 6 7 89 99 1 2 3 4 5 6 7 8 9 9 1|2 3 4 5 6 7 8 9 9 6 7 8 9 a b c d e 9 6 7 8 9 a b c de 99 1|2 3 4 5 6 789 999 1|2 3 4 5 6 7 89 99 1|2 3 4 5 6789 9999 1|2 3 4 56789 99999 1|2|3 4 5 6 7 8 9 9 1|2|3 4 5 6789 9999 1|2|3 4 5 6 789 999 1|2|3 4 5 6 7 89 99 1|2|3|4 5 6 7 89 99 1|2|3|4 5 6 789 999 1|2|3|4 5 6 7 8 9 9 6 7 8 9 a b cde 999 1|2|3|4|5 6 7 8 9 9 1|2|3|4|5 6 7 89 99 6 7 8 9 a bcde 9999 1|2|3|4|5 7 8 9 9 6 7 8 9 abcde 99999 6 7 8 9abcde e99999 6|7 8 9 a b c d e 9 6|7 8 9 a b c de 99 6|7 8 9 a b cde 999 6|7 8 9 a bcde 9999 6|7 8 9 abcde 99999 6|7|8|9|a b c d e 9 6|7|8|9|a b c de 99 6|7|8|9|a|b c d e 9 ARM on either SP1 or SP2 (W or E) does not seem to help much in identifying communities.

  20. C640 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 E040 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 D9 E1 E2 E3 E4 E5 E6 H8 H9 APPENDIX SP3 Lists 8021 22 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41 42 43 45 46 49 78 79 83 84 85 38 1 2 55 72 76 81 88 D2 H1 H6 8141 43 46 81 87 88 95 96 E146 72 73 74 81 87 95 96 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E2 E3 E4 E5 E6 42 41 43 47 48 50 76 81 88 D2 H1 C740 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 8638 81 8746 81 95 96 43 45 47 48 50 51 79 83 84 85 91 8840 41 43 46 76 81 86 88 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 E246 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E3 E4 E5 E6 C846 72 73 74 81 87 95 96 C6 C7 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 9127 43 45 47 48 50 51 77 78 79 80 83 84 85 88 91 A6 A7 A8 A9 B0 B2 C4 D2 H4 I0 E346 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E4 E5 E6 C941 43 46 81 88 E446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E5 E6 D046 72 73 74 81 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H0 H1 H2 H3 H4 H5 H6 H7 H8 9240 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 D546 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E6 G6 H5 H3 G5 H4 I0 H1 H8 H9 H6 H2 H0 G1 40 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H0 H1 H2 H3 H5 H6 H7 H9 43 46 78 79 81 H0 H1 H2 H3 H5 H7 H8 G6 46 81 H0 H2 H3 H4 H5 H6 H7 H8 46 81 H0 H1 H2 H3 H5 H7 H8 46 81 H1 H2 H3 H4 H5 H6 H7 H8 45 47 48 50 51 77 91 G1 I0 41 44 46 81 87 88 95 96 H4 46 81 H0 H1 H2 H3 H4 H5 H6 H7 H8 H4 46 76 81 H0 H1 H2 H3 H6 H7 H8 46 81 G1 H0 H1 H3 H4 H5 H6 H7 H8 44 H4 H9 9340 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 D141 43 46 72 73 74 81 88 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 9440 46 53 54 55 56 57 58 59 60 61 62 63 64 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 45 41 43 45 47 48 50 76 D2 H1 81 88 D245 47 48 49 78 79 91 E7 E8 E9 F0 46 46 49 55 72 78 B2 D1 H6 H7 D346 72 73 74 C6 C7 C8 C9 D1 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 47 46 49 D2 9546 81 87 96 48 45 48 50 51 72 74 9641 43 46 81 88 D446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 9798 99 A4 A5 49 45 47 48 49 50 51 D2 9840 46 76 81 86 89 90 97 99M A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 50 45 47 48 49 50 H4 D546 72 73 74 C6 C7 C8 C9 D1 D3 D8 D9 E0 E1 E2 E3 E4 E5 E6 9978 91 51 45 47 48 50 A498 99 A5 52 45 47 48 49 50 51 66 67 68 69 70 75 B3 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 E8 A598 99 A4 A788 91 A3 53 46 47 49 B1 76 H1 54 46 47 49 B277 G1 H4 B3 76 H1 55 46 71 74 D746 72 73 74 91 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B4B5 B6 C0 C1 C2 C3 56 46 51 55 77 B2 H6 63 1 B5B4 B6 C0 C1 C2 C3 49 46 74 C076 B4 B5 B6 C1 C2 C3 H1 72 46 48 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 D846 72 73 74 C6 C7 C8 C9 D1 D3 D5 D9 E0 E1 E2 E3 E4 E5 E6 E9 73 46 47 49 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 C177 B4 B5 B6 C0 C1 C2 C3 H5 74 46 49 71 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 C2B4 B5 B6 C0 C1 C3 D940 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 E0 E1 E2 E3 E4 E5 E6 H8 H9 7576 H5 7676 H5 C477 B2 B7 B8 B9 C4 77H4 7845 47 48 50 51 78 79 83 85 91 C5 46 51 78 B2 D1 D2 H7 7946 47 48 50 51 78 79 83 84 85

  21. C640 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 E040 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 D9 E1 E2 E3 E4 E5 E6 H8 H9 APPENDIX SP3 Lists 8021 22 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41 42 43 45 46 49 78 79 83 84 85 38 1 2 55 72 76 81 88 D2 H1 H6 8141 43 46 81 87 88 95 96 E146 72 73 74 81 87 95 96 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E2 E3 E4 E5 E6 42 41 43 47 48 50 76 81 88 D2 H1 C740 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H8 H9 8638 81 8746 81 95 96 43 45 47 48 50 51 79 83 84 85 91 8840 41 43 46 76 81 86 88 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 E246 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E3 E4 E5 E6 45 41 43 45 47 48 50 76 D2 H1 81 88 C846 72 73 74 81 87 95 96 C6 C7 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 9127 43 45 47 48 50 51 77 78 79 80 83 84 85 88 91 A6 A7 A8 A9 B0 B2 C4 D2 H4 I0 E346 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E4 E5 E6 C941 43 46 81 88 E446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E5 E6 D046 72 73 74 81 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 H0 H1 H2 H3 H4 H5 H6 H7 H8 9240 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 D546 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E6 H1 H4 I0 H8 H6 H5 H2 H3 H0 G1 G6 G5 H9 43 46 78 79 81 H0 H1 H2 H3 H5 H7 H8 40 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H0 H1 H2 H3 H5 H6 H7 H9 44 H4 H9 46 81 H0 H1 H2 H3 H4 H5 H6 H7 H8 G6 H4 46 81 H1 H2 H3 H4 H5 H6 H7 H8 45 47 48 50 51 77 91 G1 I0 41 44 46 81 87 88 95 96 H4 46 81 H0 H1 H2 H3 H5 H7 H8 46 76 81 H0 H1 H2 H3 H6 H7 H8 46 81 G1 H0 H1 H3 H4 H5 H6 H7 H8 46 81 H0 H2 H3 H4 H5 H6 H7 H8 9340 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 D141 43 46 72 73 74 81 88 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 45 46 51 78 B2 D1 H7 45 41 43 45 47 48 50 76 81 88 D2 H1 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 9440 46 53 54 55 56 57 58 59 60 61 62 63 64 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 46 46 49 55 72 78 B2 D1 H6 H7 D245 47 48 49 78 79 91 E7 E8 E9 F0 47 46 49 D2 46 45 47 48 50 51 48 45 48 50 51 72 74 D346 72 73 74 C6 C7 C8 C9 D1 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 47 48 72 48 46 47 49 9546 81 87 96 49 45 47 48 49 50 51 D2 49 46 71 74 9641 43 46 81 88 D446 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 50 45 47 48 49 50 H4 9798 99 A4 A5 50 46 51 55 H6 9840 46 76 81 86 89 90 97 99M A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 51 46 52 46 53 54 55 56 57 58 59 60 61 62 63 64 D546 72 73 74 C6 C7 C8 C9 D1 D3 D8 D9 E0 E1 E2 E3 E4 E5 E6 9978 91 53 48 51 45 47 48 50 A498 99 A5 54 48 52 45 47 48 49 50 51 66 67 68 69 70 75 B3 D646 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 E8 A598 99 A4 55 49 A788 91 A3 53 46 47 49 B1 76 H1 56 50 75 B3 54 46 47 49 B277 G1 H4 57 69 B3 76 H1 55 46 71 74 D746 72 73 74 91 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 B4B5 B6 C0 C1 C2 C3 58 70 56 46 51 55 77 B2 H6 63 1 59 68 B5B4 B6 C0 C1 C2 C3 49 46 74 60 67 C076 B4 B5 B6 C1 C2 C3 H1 72 46 48 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 61 66 D846 72 73 74 C6 C7 C8 C9 D1 D3 D5 D9 E0 E1 E2 E3 E4 E5 E6 E9 63 66 73 46 47 49 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 C177 B4 B5 B6 C0 C1 C2 C3 H5 65 E7 74 46 49 71 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 C2B4 B5 B6 C0 C1 C3 D940 46 72 73 74 76 81 86 89 90 97 98 99 A0 A1 A2 A3 A4 B1 B2 C6 C7 C8 C9 D1 D3 D5 D8 E0 E1 E2 E3 E4 E5 E6 H8 H9 66 61 7576 H5 71 49 7676 H5 C477 B2 B7 B8 B9 C4 77H4 72 47 D2 7845 47 48 50 51 78 79 83 85 91 C5 46 51 78 B2 D1 D2 H7 73 48 D2 7946 47 48 50 51 78 79 83 84 85 74 49 D2

  22. 18 1 7577 7677 19 2 7776 H5 20 3 7841 43 46 81 88 21 4 22 5 7946 81 87 95 96 23 6 8038 81 24 7 8178 79 83 84 85 25 8 D246 72 73 74 C6 C7 C8 C9 D1 D3 D5 D8 D9 E0 E1 E2 E3 E4 E5 E6 26 9 8680 D3D2 27 10 8779 28 11 D4D2 8878 91 29 12 D5D2 30 13 8985 31 14 90A6 32 15 9140 46 76 81 86 89 90 97 98 99 A0 A1 A2 A4 B1 B4 C6 C7 H8 H9 33 16 9291 34 17 9391 35 18 36 19 9491 52 D6E5 D2 37 20 23 38 21 22 24 25 26 27 28 29 30 31 32 33 34 35 36 40 41 42 43 45 46 49 D7D9 D2 39 29 D8E4 D2 40 27 9579 D991 D2 42 45 F6 G7 G6 G5 G4 H1 H2 H3 G1 H5 H4 H6 F7 H8 H0 H9 I0 G3 G2 G1 H4 F8 F7 H0 H4 H1 F6 44 H4 H9 F7 H4 I0 G0 77 H4 H4 I0 43 78 79 46 81 H0 H1 H2 H3 H5 H6 H7 H8 F9 91 H4 G1 H4 G6 G7 43 78 9678 E091 D2 97A7 45 46 51 78 B2 D1 H7 E179 D2 9891 E2D2 46 45 47 48 50 51 9988 E3F0 D2 47 48 72 A0A8 E4E9 D2 48 46 47 49 A1A9 E5E8 D2 49 46 71 74 A2B0 E6E7 D2 50 46 51 55 H6 A4A7 51 46 A5A3 A7 52 46 53 54 55 56 57 58 59 60 61 62 63 64 A798 99 A4 A5 53 48 54 48 B1 B2 B276 H1 55 49 B3B2 56 50 75 B3 B4C4 57 69 B5C4 58 70 B6B7 59 68 C0B2 C4 60 67 C177 C4 61 66 C2B9 C4 63 66 C3B8 65 E7 C4B4 B5 B6 C0 C1 C2 C3 66 61 71 49 c5 45 D5 72 47 D2 C691 D2 73 48 D2 C791 D2 74 49 D2 C879 D2 C978 D0H4 D2 D178 D2

  23. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 End 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 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 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 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 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 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 Strt 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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

  24. G10 (v=1-44) SP2 2level pTrees LevelOneStride=19 (labelled 0-i), Level0Stride=10 (labelled 0-9) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 221 1 1 1 OutDegSP1 1 8 1 0 0 0 0 1 9 1 0 0 0 0 2 0 1 0 0 0 0 2 1 1 0 0 0 0 2 2 1 0 0 0 0 2 3 1 0 0 0 0 2 4 1 0 0 0 0 2 5 1 0 0 0 0 2 6 1 0 0 0 0 2 7 0 1 0 0 0 2 8 0 1 0 0 0 2 9 0 1 0 0 0 3 0 0 1 0 0 0 3 1 0 1 0 0 0 3 2 0 1 0 0 0 3 3 0 1 0 0 0 3 4 0 1 0 0 0 3 5 0 1 0 0 0 3 6 0 1 0 0 0 3 7 0 0 1 0 0 3 8 0 0 1 1 1 3 9 0 0 1 0 0 4 0 0 0 1 0 0 4 2 0 0 0 0 1 4 3 0 0 0 0 0 0 0 1 0 0 L1 0 1 2 3 4 5 6 7 8 9 L0 0 1 2 3 4 5 6 7 8 9 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 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 0 0 1 0 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 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 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 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 1 1 1 1 1 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 1 0 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 L0 0 1 2 3 4 5 6 7 8 9 1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 2 211 1 1 1 OutDegSP2 3 5 1 0 0 0 0 3 8 0 0 0 0 0 1 0 1 0 0 4 0 0 0 1 0 0 4 2 0 0 0 0 1 4 3 0 0 0 0 0 0 0 1 0 0 3 6 1 0 0 0 0 3 7 1 0 0 0 0 L1 0 1 2 3 4 5 6 7 8 9 A B C D E F G H i L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 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 1 0 0 0 L0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 1 0 0 0 1 1 112 131 177 G10 leaves (OutDegree=0): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 41 44 62 64 67 68 69 70 82 83 84 85 a3 a6 a8 a9 b0 B7 b8 b9 e7 e8 e9 f0 f1 f2 f3 f4 f5 f8 f9 g0 g8 g9 h7

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