1. 1 Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green Flexible Models for Complex Networks

2. The Internet is constantly growing and evolving giving rise to new models and algorithmic questions. 2 degree 4102 100 frequency, but no sharp concentration: Erdos-Renyi Sparse graphs with large degree-variance. “Power-law” degree distributions. Small-world, i.e. small diameter, high clustering coefficients.

3. degree 4102 100 frequency, but no sharp concentration: Erdos-Renyi Sparse graphs with large degree-variance. “Power-law” degree distributions. Small-world, i.e. small diameter, high clustering coefficients. A rich theory of power-law random graphs has been developed [ Evolutionary & Configurational Models, e.g. see Rick Durrett’s ’07 book ]. However, in practice, there are discrepancies …

4. 4 “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.

5. 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). Models with semantics on nodes, and links among nodes with semantic proximity generated by very general probability distributions. RANDOM DOT PRODUCT GRAPHS KRONECKER GRAPHS Talk Outline1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Generalizations of Erdos-Gallai / Havel-Hakimi Generalizations of Erdos-Renyi random graphs

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). Talk Outline 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. Assortativity: smalllarge

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

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

9. 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. The Joint-Degree Matrix Realization Problem is: Open: Mincost, Random realizations of

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

11. 11 Theorem [ Erdos-Gallai ]: A degree sequence is realizable iff the natural necessary condition holds: moreover, there is a connected realization iff the natural necessary condition holds:

12. 12 4 3 2 2 1 1 1 [ Havel-Hakimi ] Construction Algorithm : Greedy : any unsatisfied vertex is connected with the vertices of highest remaining degree requirements. 0 3 2 2 0 0 0 1 0 1 0 0 Connectivity, if possible, attained with 2-switches. add delete add delete

13. Random generation of graph with a given degree sequence: Theorem [Cooper, Frieze & Greenhill 04]:The Markov chain corresponding to a general 2-link switch is rapidly mixing for degree sequences with.

14. 14 Random generation of connected graph with a given degree sequence: 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.

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

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

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

18. Main Difficulty: Two connected components are amenable to rewiring by 2-switches, only using two vertices of the same degree. connected component connected component The algorithm explores vertices of the same degree in different components, transforming the graph to bring it to a form amenable to rewiring by 2-switch, if possible.

19. Certificates of non-existence of connected realizations result from contractions of subsets of performed by the algorithm (as it was searching for transformations amenable to 2-switch rewirings across connected components.) 1 1 1 2 1 4 2 2 4 3 2 9 available edges & 11 vertices. There are not enough edges to connect all the vertices!

20.  20 Open Problems for Joint Degree Matrix Realization  Construct mincost realization.  Construct random realization.  Satisfy constraints between arbitrary subsets of vertices.  Is there a reduction to matching or flow or some other well understood combinatorial problem?  Is there evidence of hardness?

21. Models with semantics on nodes, and links among nodes with semantic proximity generated by very general probability distributions. RANDOM DOT PRODUCT GRAPHS KRONECKER GRAPHS Talk Outline1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Generalizations of Erdos-Gallai / Havel-Hakimi Generalizations of Erdos-Renyi random graphs

22. RANDOM DOT PRODUCT GRAPHS Kratzl,Mickel,Sheinerman 05 Young,Sheinerman 07 Young,M 08

23. SUMMARY OF RESULTS  A semi-closed formula for degree distribution and graphs with a wide variety of densities and degree distributions, including power-laws.  Diameter characterization (determined by Erdos-Renyi for similar average density)  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.

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

25. Example: (a wide range of degrees, except for very large degrees) indicating a power-law with exponent between 2 and 3. This is in agreement with real data.

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

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

28. 28 Open Problems for Random Dot Product Graphs  Fit real data, and isolate “benchmark” distributions.  Characterize connectivity (diameter and conductance) as approaches.  Similarity functions beyond inner product (e.g. Kernel functions).  Algorithms: navigability, information/virus propagation, etc.  Do further properties of X characterize further properties of ?

29. KRONECKER GRAPHS [Faloutsos, Kleinberg,Leskovec 06] 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 Another “semantic” “ flexible” model, introducing parametrization. Several properties characterized.

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

31. Summary1. Structural/Syntactic Flexible Models 2. Semantic Flexible Models Generalizations of Erdos-Gallai / Havel-Hakimi Generalizations of Erdos-Renyi random graphs

32. 32 Where it all started: Kleinberg’s navigability model Theorem [Kleinberg]: The only value for which the network is navigable is r =2. Parametrization is essential in the study of complex networks Moral: ?