Fan Chung Graham University of California, San Diego

1. New Directions in Graph Theory for network sciences Fan Chung Graham University of California, San Diego

2. A graph G = (V,E) edge vertex

3. Vertices cities people authors telephones web pages genes Edges flights pairs of friends coauthorship phone calls linkings regulatory aspects Graph models _____________________________

4. Graph Theory has 250 years of history. Leonhard Euler 1707-1783 The bridges of Königsburg Is it possible to walk over every bridge once and only once?

5. Graph Theory has 250 years of history. Theory applications Real world large graphs

6. Geometric graphs Algebraic graphs real graphs

7. Massive data Massive graphs • WWW-graphs • Call graphs • Acquaintance graphs • Graphs from any data a.base

8. The Opte project

9. An Internet routing (BGP) graph

10. A subgraph of the Hollywood graph.

11. An induced subgraph of the collaboration graph with authors of Erdös number ≤ 2.

12. Numerous questions arise in dealing with large realistic networks • How are these graphs formed? • What are the basic structures of such xxgraphs? • What principles dictate their behavior? • How are subgraphs related to the large xxhost graph? • What are the main graph invariants xxcapturing the properties of such graphs?

13. New problems and directions • Classical random graph theory Random graphs with any given degrees • Percolation on special graphs Percolation on general graphs • Correlation among vertices Pagerank of a graph • Graph coloring/routing Network games

14. Several examples • Random graphs with specified degrees Diameter of random power law graphs • Diameter of random trees of a given graph • Percolation and giant components in a graph • Correlation between vertices xxxxxxxxxxxxThe pagerank of a graph • Graph coloring and network games

15. Classical random graphs Same expected degree for all vertices Random graphs with specified degrees Random power law graphs

16. Some prevailing characteristics of large realistic networks • Sparse • Small world phenomenon Small diameter/average distance Clustering • Power law degree distribution

17. 3 3 4 4 edge 2 4 vertex Degree sequence: (4,4,4,3,3,2) Degree distribution: (0,0,1,2,3)

18. A crucial observation Discovered by several groups independently. • Broder, Kleinberg, Kumar, Raghavan, Rajagopalan aaand Tomkins, 1999. • Barabási, Albert and Jeung, 1999. • M Faloutsos, P. Faloutsos and C. Faloutsos, 1999. • Abello, Buchsbaum, Reeds and Westbrook, 1999. • Aiello, Chung and Lu, 1999. Massive graphs satisfy the power law.

19. The history of the power law • Zipf’s law, 1949. (The nth most frequent word occurs at rate 1/n) • Yule’s law, 1942. • Lotka’s law, 1926. (Distribution of authors in chemical abstracts) • Pareto, 1897 (Wealth distribution follows a power law.) 1907-1916 (City populations follow a power law.) Natural language Bibliometrics Social sciences Nature

20. Power law graphs Power decay degree distribution. The degree sequences satisfy apower law: The number of vertices of degree j is proportional to j-ß where ß is some constant ≥ 1.

21. Comparisons From real data From simulation

22. The distribution of the connected componentsin the Collaboration graph

23. The distribution of the connected componentsin the Collaboration graph The giant component

24. Examples of power law • Inter • Internet graphs. • Call graphs. • Collaboration graphs. • Acquaintance graphs. • Language usage • Transportation networks

25. Degree distribution of an Internet graph A power law graph with β = 2.2 Faloutsos et al ‘99

26. Degree distribution of Call Graphs A power law graph with β = 2.1

27. The collaboration graph is a power law graph, based on data from Math Reviews with 337451 authors A power law graph with β = 2.25

28. The Collaboration graph (Math Reviews) • 337,000 authors • 496,000 edges • Average 5.65 collaborations per person • Average 2.94 collaborators per person • Maximum degree 1416 • The giant component of size 208,000 • 84,000 isolated vertices (Guess who?)

29. What is the `shape’ of a network ? experimental modeling

30. Massive Graphs Random graphs Similarities: Adding one (random) edge at a time. Differences: Random graphs almost regular. Massive graphs uneven degrees, correlations.

31. Random Graph Theory How does a random graph behave? Graph Ramsey Theory What are the unavoidable patterns?

32. Paul ErdÖs and A. Rényi, On the evolution of random graphs Magyar Tud. Akad. Mat. Kut. Int. Kozl. 5 (1960) 17-61.

33. A random graph G(n,p) • G has n vertices. • For any two vertices u and v in G, a{u,v} is anedge with probability p.

34. What does a random graph look like?

35. Prob(G is connected)?

36. no. of connected graphs Prob(G is connected) = total no. of graphs

37. A random graph has property P Prob(G has property P) as

38. Random graphs with expected degrees wi wi :expected degree at vi Prob( i ~ j) = wiwj p Choose p = 1/wi , assuming max wi2< wi. Erdos-Rényi model G(n,p) : The special case with same wi for all i.

39. Small world phenomenon Six degrees of separation Milgram 1967 Two web pages (in a certain portion of the Web) are 19 clicks away from each other. / 39 Barabasi 1999 Broder 2000

40. Distance d(u,v) = length of a shortest path joining u and v. Diameter diam(G) = max { d(u,v)}. u,v Average distance = ∑ d(u,v)/n2. u,v where u and v are joined by a path.

41. Exponents for Large NetworksP(k)~k -

42. Properties of Chung+Lu PNAS’02 Random power law graphs > 3 average distance diameter c log n log n / log = 3 average distance log n / log log n diameter c log n 2 << 3 average distance log log n diameter c log n provided d > 1 and max deg `large’

43. The structure of random power law graphs 2 << 3 `Octopus’ Core has width log log n core legs of length log n

44. Yahoo IM graph

45. Several examples • Random graphs with any given degrees Diameter of random power law graphs • Diameter of random trees of a given graph • Percolation and giant components in a graph • Correlation between vertices xxxThe pagerank of a graphs • Graph coloring and network games

46. Motivation 2008

47. Motivation Random spanning trees have large diameters.

48. Diameter of spanning trees Theorem (Rényi and Szekeres 1967): The diameter of a random spanning tree in a complete graph Kn is of order . Theorem (Aldous 1990) : The diameter diam(T) of a random spanning tree in a regular graph with spectral bound  is

49. The spectrum of a graph Many ways to define the spectrum of a graph Adjacency matrix How are the eigenvalues related to properties of graphs?

50. The spectrum of a graph • Adjacency matrix • Combinatorial Laplacian adjacency matrix diagonal degree matrix • Normalized Laplacian Random walks Rate of convergence