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Milena Mihail Georgia Tech.

“with categorical attributes”. [C. Faloutsos MMDS08]. Models and Algorithms for Complex Networks. “with network elements maintaining characteristic profiles ”. “with parametrization , typically local ”. Milena Mihail Georgia Tech. with

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Milena Mihail Georgia Tech.

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  1. “with categorical attributes” [C. Faloutsos MMDS08] Models and Algorithms for Complex Networks “with network elements maintaining characteristic profiles” “with parametrization, typically local” Milena Mihail Georgia Tech. with Stephen Young, Gagan Goel, Giorgos Amanatidis, Bradley Green, Christos Gkantsidis, Amin Saberi, D. Bader, T. Feder, C. Papadimitriou, P. Tetali, E. Zegura

  2. Complex Networks in Web N.0 Talk Outline Flexible (further parametrized) Models 1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Models & Algorithms Connection : Kleinberg’s Model(s) for Navigation Distributed Searching Algorithms with Additional Local Info/Dynamics 1. On the Power of Local Replication 2. On the Power of Topology Awareness via Link Criticality Conclusion : Web N.0 Model & Algorithm characteristics: further parametrization, typically local, locality of info in algorithms & dynamics. Dynamics become especially important. 2

  3. , but no sharp concentration: frequency Erdos-Renyi 100 2 4 10 degree Sparse graphs with large degree-variance. “Power-law” degree distributions. Small-world, i.e. small diameter, high clustering coefficients. scaling Web N.0 The Internet is constantly growing and evolving giving rise to new models and algorithmic questions.

  4. However, in practice, there are discrepancies … , but no sharp concentration: frequency Erdos-Renyi 100 2 4 10 degree Sparse graphs with large degree-variance. “Power-law” degree distributions. Friendship Network Flickr Network Patent Collaboration Network (in Boston) Small-world, i.e. small diameter, high clustering coefficients. Random Graph with same degrees Global Routing Network (Autonomous Systems) Local Routing Network with same Degrees A rich theory of power-law random graphs has been developed [ Evolutionary, Configurational Models, & e.g. see Rick Durrett’s ’07 book ]. 4

  5. “Flexible” models for complex networks: exhibit a “large” increase in the properties of generated graphs by introducing a “small” extension in the parameters of the generating model.

  6. Models with power law and arbitrary degree sequences with additional constraints, such as specified joint degree distributions (from random graphs, to graphs with very low entropy). Assortativity: Case 1: Structural/Syntactic Flexible Model Modifications and Generalizations of Erdos-Gallai / Havel-Hakimi small large The networking community proposed that [Sigcomm 04, CCR 06 and Sigcomm 06], beyond the degree sequence , models for networks of routers should capture how many nodes of degree are connected to nodes of degree . 6

  7. Networking Proposition [CCR 06, Sigcomm 06]: A random graph with same average degree as G. A real highly optimized network G. A random graph with same degree sequence as G. A graph with same number of links between nodes of degree and as G. 7 A graph with same as G. A random graph with same degree sequence as G. A graph with same as G. A random graph with same degree sequence as G. A random graph with same degree sequence as G. A random graph with same degree sequence as G. A random graph with same degree sequence as G. A random graph with same degree sequence as G.

  8. The Joint-Degree Matrix Realization Problem is: connected, mincost, random connected, mincost, random Definitions The (well studied) Degree Sequence Realization Problem is: 8

  9. The Joint-Degree Matrix Realization Problem is: connected, mincost, random Theorem [Amanatidis, Green, M ‘08]: The natural necessary conditions for an instance to have a realization are also sufficient (and have a short description). The natural necessary conditions for an instance to have a connected realization are also sufficient (no known short description). There are polynomial time algorithms to construct a realization and a connected realization of , or produce a certificate that such a realization does not exist. 9

  10. Degree Sequence Realization Problem: Given arbitrary Is this degree sequence realizable ? If so, construct a realization. Advantages:Flexibilityin: enforcing or precluding certain edges, adding costs on edges and finding mincost realizations, close to matching  close to sampling/random generation. connected, mincost, random Reduction to perfect matching: 2

  11. Theorem[Erdos-Gallai]: A degree sequence is realizable iff the natural necessarycondition holds: moreover, there is a connectedrealization iff the natural necessary condition holds: 1 1 4 1 3 2 2 0 0 0 1 2 3 0 [ Havel-Hakimi ] Construction: Greedy: any unsatisfied vertex is connected with the vertices of highest remaining degree requirements. 0 1 2 0 Connectivity, if possible, attained with 2-switches . 0 Note :all 2-switches are legal . 11

  12. Theorem, Joint Degree Matrix Realization[Amanatidis, Green, M ‘08]: Proof [sketch]: 12

  13. Note:This may NOT be a simple “augmenting” path. Balanced Degree Invariant: Example Case Maintaining Balanced Degree Invariant: add add delete delete add

  14. Theorem, Joint Degree Matrix Connected Realization [Amanatidis, Green, M ‘08]: Proof [remarks]: Main Difficulty: Two connected components are amenable to rewiring by 2-switches, only using two vertices of the same degree. The algorithm explores vertices of the same degree in different components, in a recursive manner. connected component connected component 14

  15. Open ProblemsforJoint Degree Matrix Realization • Construct mincost and/or random realization, or connected realization. • Satisfy constraints between arbitrary subsets of vertices. • Is there a reduction to matchings or flow or some other well understood combinatorial problem? • Is there evidence of hardness ? • Is there a simple generative model for graphs with low assortativity ? ( explanatory or other …) 15

  16. Complex Networks in Web N.0 Talk Outline Flexible (further parametrized) Models 1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Models & Algorithms Connection : Kleinberg’s Model(s) for Navigation Distributed Searching Algorithms with Additional Local Info/Dynamics 1. On the Power of Local Replication 2. On the Power of Topology Awareness via Link Criticality Conclusion : Web N.0 Model & Algorithm characteristics: further parametrization, typically local, locality of info in algorithms & dynamics. Dynamics become especially important. ODD

  17. Models with semantics on nodes, and links among nodes with semantic proximity generated by verygeneral probability distributions. Friendship Network Flickr Network Patent Collaboration Network (in Boston) • RANDOM DOT PRODUCT GRAPHS • KRONECKER GRAPHS Varying structural characteristics Also densification, shrinking diameter, … C. Faloutsos, MMDS08 Case 2: Semantic Flexible Model(s) Generalizations of Erdos-Renyi random graphs 16

  18. RANDOM DOT PRODUCT GRAPHS Kratzl,Nickel,Scheinerman 05 Young,Scheinerman 07 Young,M 08 17

  19. very likely never somewhat likely unlikely 18

  20. SUMMARY OF RESULTS • A semi-closed formula for degree distribution Model can generate graphs with a wide variety of densities average degrees up to . and wide varieties of degree distributions, including power-laws. • Diameter characterization : Determined by Erdos-Renyi for similar average density, if all coordinates of X are in (will say more about this). • Positive clustering coefficient, depending on the “distance” of the generating distribution from the uniform distribution. Remark: Power-laws and the small world phenomenon are the hallmark of complex networks. 19

  21. A Semi-closed Formula for Degree Distribution Theorem [Young, M ’08] Theorem ( removing error terms) [Young, M ’08] 20

  22. Example: (a wide range of degrees, except for very large degrees) This is in agreement with real data. 21

  23. Theorem [Young, M ’08] Diameter Characterization Remark: If the graph can become disconnected. It is important to obtain characterizations of connectivity as approaches .This would enhance model flexibility 22

  24. Clustering Characterization Theorem [Young, M ’08] Remarks on the proof 23

  25. Open Problems forRandom Dot Product Graphs • Fit real data, and isolate “benchmark” distributions . • Characterize connectivity (diameter and conductance) as approaches . • Do/which further properties of X characterizefurther properties of ? • Evolution:X as a function of n ? (including: two connected vertices with small similarity, either disconnect, or increase their similarity). Should/can ? • Similarity functions beyond inner product (e.g. Kernel functions). • Algorithms: navigability, information/virus propagation, etc.

  26. 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 KRONECKER GRAPHS [Faloutsos, Kleinberg,Leskovec 06] 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 1 2-bit vertex character- ization 1-bit vertex character- ization 0 0 1 1 0 0 1 1 3-bit vertex character- ization 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 Another “semantic” “ flexible” model, introducing parametrization. 25

  27. a b aaa baa aa caa baa bab bab aab ab cab aba bba ba cba bba bbb bb bbb abb cbb b c aab bab bab ab cab bac ac bac aac cac bbb abb bb cbb bbb bbc bbc abc cbc bc cba bba bba aba ba bbb bb bbb abb cbb bca ca cca bca bcb acb bcb ccb cb cbb bbb bbb abb bb bbc abc bc cbc bbc ccd acd bcd cd bcd acc bcc bcc ccc cc STOCHASTIC KRONECKER GRAPHS [ Faloutsos, Kleinberg, Leskovec 06] aca Several properties characterized (e.g. multinomial degree distributions, densification, shrinking diameter, self-similarity). Large scale data set have been fit efficiently ! 26

  28. Complex Networks in Web N.0 Talk Outline Flexible (further parametrized) Models 1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Models & Algorithms Connection : Kleinberg’s Model(s) for Navigation Distributed Searching Algorithms with Additional Local Info/Dynamics 1. On the Power of Local Replication 2. On the Power of Topology Awareness via Link Criticality Conclusion : Web N.0 Model & Algorithm characteristics: Further Parametrization, Locality of Info & Dynamics. 27

  29. Where it all started: Kleinberg’s navigability model ? Moral: Parametrization is essential in the study of complex networks Theorem [Kleinberg]: The only value for which the network is navigable isr =2.

  30. Strategic Network Formation Process [Sandberg 05]: Experimentally, the resulting network has structure and navigability similar to Kleinberg’s small world network. 30

  31. Strategic Network Formation Process [ Green & M ’08 ] simplification of [Clauset & Moore 03]: Experimentally, the resulting network becomes navigable after steps but does not have structure similar to Kleinberg’s small world network. 30

  32. Complex Networks in Web N.0 Talk Outline Flexible (further parametrized) Models 1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Models & Algorithms Connection : Kleinberg’s Model(s) for Navigation Distributed Searching Algorithms with Additional Local Info/Dynamics 1. On the Power of Local Replication 2. On the Power of Topology Awareness via Link Criticality Conclusion : Web N.0 Model & Algorithm characteristics: further parametrization, typically local, locality of info in algorithms & dynamics. Dynamics become especially important. 31

  33. 1. On the Power of Local Replication How do networks search (propagate information) : [ Gkantidis, M, Saberi , ‘04 ‘05 ] Cost = queried nodes / found information 32

  34. Theorem (extends to power-law random graphs): Proof : 1. On the Power of Local Replication [ Gkantidis, M, Saberi , ‘04 ‘05 ] [M, Saberi , Tetali ‘05 ] Equivalent to one-step local replication of information. 33

  35. 2. On the Power of Topology Awareness via Link Criticality 34

  36. Via Semidefinite Programming “Fastest Mixing Markov Chain” Boyd,Diaconis,Xiao 04 Via Computation of Principal Eigenvector(s), Distributed, Asynchronous Gkantsidis, Goel, Mihail, Saberi 07 2. On the Power of Topology Awareness via Link Criticality How do social networks compute link criticality ? 35

  37. + + + + - - - - - + Start with Step: For all nodes Step: For links, asynchronously Link Criticality via Distributed Asynchronous Computation of Principal Eigenvector(s) [Gkantsidis,Goel,M,Saberi 07] +1 +1 -1 -1 Hardest part: Numerical Stability. 36

  38. Complex Networks in Web N.0 Talk Outline Flexible (further parametrized) Models 1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Models & Algorithms Connection : Kleinberg’s Model(s) for Navigation Distributed Searching Algorithms with Additional Local Info/Dynamics 1. On the Power of Local Replication 2. On the Power of Topology Awareness via Link Criticality Conclusion : Web N.0 Model & Algorithm characteristics: further parametrization, typically local, locality of info in algorithms & dynamics. Dynamics become especially important. 37

  39. Theorem[Feder,Guetz,M,Saberi 06]:The Markov chain corresponding to a local 2-link switch is rapidly mixing if the degree sequence enforces diameter at least 3, and for some . Topology Maintenance = Connectivity & Good Conductance 39

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