Fan Chung Graham University of California, San Diego

A graph G = (V,E) vertex edge

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

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

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

Geometric graphs Algebraic graphs real graphs

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

The Opte project

An Internet routing (BGP) graph

A subgraph of the Hollywood graph.

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

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

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

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

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

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

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

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

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

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

Comparisons From simulationFrom real data

The distribution of the connected components in the Collaboration graph

The giant component

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

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

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

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

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?)

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

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

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

Paul Erd Ö s and A. Rényi, On the evolution of random graphs Magyar Tud. Akad. Mat. Kut. Int. Kozl. 5 (1960)

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

What does a random graph look like?

Prob( G is connected)?

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

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

w i : expected degree at v i Random graphs with expected degrees w i Prob( i ~ j ) = w i w j p Erdos-Rényi model G(n,p) : The special case with same w i for all i. Choose p = 1/  w i, assuming max w i 2 <  w i.

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

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)/n 2. u,v where u and v are joined by a path.

Exponents for Large Networks P(k)~k -  NetworksWWWActorsCitation Index Power Grid Phone calls  ~2.1 (in) ~2.5 (out) ~2.3~3~4~2.1

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

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

Yahoo IM graph

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

Motivation 2008

Motivation Random spanning trees have large diameters.

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

Adjacency matrixMany ways to define the spectrum of a graph How are the eigenvalues related to properties of graphs? properties of graphs? The spectrum of a graph

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

For a path Discrete Laplace operator ∆ on f: V  R The spectrum of a graph

not symmetric in general Normalized Laplacian symmetric normalized {{ Discrete Laplace operator ∆ on f: V  R

Properties of Laplacian eigenvalues of a graph Spectral bound  : “ = “ holds iff G is disconnceted or bipartite.

Question What is the diameter of a random spanning tree of a given graph G ?

Some notation For a given graph G, n: the number of vertices, d x : the degree of vertex x, vol(G)=∑ x d x : the volume of G, d =vol(G)/n : the average degree, The second-order average degree  : the minimum degree,

Diameter of random spanning trees Chung, Horn and Lu 2008 If then with probability 1- , a random tree T in G has diameter diam( T ) satisfying Ifthen

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

A disease contact graph Jim Walker 2008

For a given graph G, retain each edge with probability p. Contact graph infection rate Percolation on G = a random subgraph of G. G p : Example: G=K n, G(n,p), Erdös-Rényi model Question: For what p, does G p have a giant xxxxxxxxxcomponent? Under what conditions will the disease spread to a large population?

Hammersley 1957, Fisher 1964 …… Percolation on graphs Erdös-Rényi 1959 History: Percolation on lattices d -regular expander graphs Ajtai, Komlos, Szemerédi 1982 hypercubes Cayley graphs Malon, Pak 2002 Bollobás et. al Frieze et. al dense graphs complete graphs Alon et. al. 2004

Percolation on general sparse graphs Percolation on special graphs or dense graphs

Percolation on general sparse graphs Theorem (Chung,Horn,Lu 2008) For a graph G, the critical probability for percolation graph G p is provided that the maximum degree of ∆ satisfies under some mild conditions.

Percolation on general sparse graphs Theorem (Chung+Horn +Lu) For a graph G, the percolation graph G p contains a giant component with volume provided that the maximum degree of ∆ satisfies under some mild conditions. Questions: Tighten the bounds? Double jumps?

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

What is PageRank? PageRank is a well-defined operator on any given graph, introduced by Sergey Brin and Larry Page of Google in a paper of Answer #1: Answer #2: PageRank denotes quantitative correlation between pairs of vertices. See slices of last year’s talk at

What does a sweep of PageRank look like?

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

Michael Kearns’ experiments on coloring games 2006

Michael Kearns’ experiments on coloring games 2006

Coloring graphs in a greedy and selfish way Classical graph coloring Chromatic graph theory Coloring games on graphs

Applications of graph coloring games dynamics of social networks conflict resolution Internet economics on-line optimization + scheduling

A graph coloring game At each round, each player (vertex) chooses a color randomly from a set of colors unused by his/her neighbors. Best response myopic strategy Arcante, Jahari, Mannor 2008 Nash equilibrium: Each vertex has a different color from its neighbors. Question: How many rounds does it take to converge to Nash equilibrium?

A graph coloring game Theorem (Chaudhuri,Chung,Jamall 2008) ∆ : the maximum degree of G If ∆+2 colors are available, the coloring game converges in O(log n) rounds. If ∆+1 colors are available, the coloring game may not converge for some initial settings.

Improving existing methods Probabilistic methods, random graphs. Random walks and the convergence rate Lower bound techniques General Martingale methods Geometric methods Spectral methods

New directions in graph theory Diameter of random trees of a given graph Random graphs with any given degrees Diameter of random power law graphs Percolation and giant components in a graph Correlation between vertices xxxThe pagerank of a graphs Graph coloring and network games Many new directions and tools ….