Fan Chung Graham University of California, San Diego

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 1707-

1783

The bridges of KönigsburgIs 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 dataMassive 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 distanceClustering

• Power law degree distribution

•Sparse

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

0

1

2

3

4

5

degree_0 degree_1 degree_2 degree_3 degree_4

vertex

edge

44

4 2

33

A crucial observation

Massive graphs satisfy the power lawpower law.

• 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.

Discovered by several groups independently.

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

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 simulation

From real data

The distribution of the connected components

in the Collaboration graph

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 GraphsA power law graph with β = 2.1

The collaboration graph is a power law graph, basedon data from Math Reviews with 337451 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, correlatio

ns.

Random Graph Theory

Graph Graph Ramsey Theory

How does a random graph How does a random graph behave?behave?

What are the What are the unavoidable patterns?unavoidable patterns?

Paul ErdÖs and A. Rényi,

On the evolution of random graphs

Magyar Tud. Akad. Mat. Kut. Int. Kozl. 5 (1960) 17-61.

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

=

48

=12

p =

12

A random graph has property P

Prob(G has property P)

as n → ∞.

→ 1

wi : expected degree at vi

Random graphs with expected degrees wi

Prob( i ~ j) = wiwj p

Erdos-Rényi model G(n,p) :

The special case with same wi for all i.

Choose p = 1/wi , assuming max wi2<

wi .

Six degrees of separationMilgram 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

Distanced(u,v) = length of a shortest path joining u and v.Diameterdiam(G) = max { d(u,v)}.

u,v

Average distance = ∑ d(u,v)/n2.

u,v

where u and v are joined by a path.

Exponents for Large Networks

P(k)~k -

Networks WWW Actors Citation 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

%d

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


• 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 Kn is of order .

Theorem (Aldous 1990) : The diameter diam(T) of a random spanning tree in a regular graph with spectral bound is

n

c(1− ) nlogn

≤E(diam(T)) ≤c nlogn

1−.

Adjacency matrixMany ways

to define

the spectrum of a graph

How are the eigenvalues How are the eigenvalues related to related to

properties of graphs?properties of graphs?

The spectrum of a graph

•Combinatorial Laplacian

L D A= −diagonal degree matrix

adjacency matrix

•Adjacency matrix

•Normalized Laplacian

Random walks

Rate of convergence


For a path

=−12{( f (xj+1) − f (xj ))

−( f (xj ) − f (xj−1))}

1( ) ( ( ) ( ))

y xx

f x f x f yd

Δ = −∑:

Discrete Laplace operator ∆ on f: V R

Δf (x)

2

2( )f x

x

∂−

∂ 1( ) ( ){ }j jf x f xx x+∂ ∂

− −∂ ∂



not symmetric in general

•Normalized Laplaciansymmetricnormalized

1( ) ( ( ) ( ))

y xx

f x f x f yd

Δ = −∑:

( , )L x y =1 if x y=

{ 1

x

if x yd

− ≠

L( , )x y =1 if x y=

{ 1

x y

if x yd d

− ≠

Discrete Laplace operator ∆ on f: V R

Properties of Laplacian eigenvalues of a graph

Spectral bound : 0 =λ0 ≤λ1 ≤⋅⋅⋅≤λn−1 ≤2

=max

i≠0|λi −1|

“=“ holds iff G is disconnceted or bipartite.

≤1

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,

• dx: the degree of vertex x,

• vol(G)=∑x dx : the volume of G,

• d =vol(G)/n : the average degree,

• The second-order average degree

%d =dx

2x∑dxx∑

• : the minimum degree,

Diameter of random spanning trees

Chung, Horn and Lu 2008

If d >>

log2 nlog2

,

then with probability 1-, a random tree T in G has diameter diam(T) satisfying

(1−)

nd%d

≤diam(T) ≤c

nd log(1 / )

logn.

If %d ≤Cd, then

Ω( n) ≤E(diam(T)) ≤O( nlogn).

Several examples



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.

Gp :

Example: G=Kn, G(n,p), Erdös-Rényi model

Question: For what p, does Gp 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 graphsMalon, Pak 2002

Bollobás et. al. 2008

Frieze et. al. 2004

• dense graphs• complete graphs

Alon et. al. 2004

Percolation on general sparse graphs

Percolation on special graphs or dense graphs


Theorem (Chung,Horn,Lu 2008)

For a graph G, the critical probability for percolation graph Gp is

p =1%d

provided that the maximum degree of ∆

satisfies

Δ =o

%d

⎛

⎝⎜⎞

⎠⎟

under some mild conditions.


Theorem (Chung+Horn +Lu)

For a graph G, the percolation graph Gp contains a giant component with volume

p =

1%d,

provided that the maximum degree of ∆

satisfies

Δ =o

%d

⎛

⎝⎜⎞

⎠⎟under some mild conditions.

max(2d log n, Ω( vol G)

Questions: Tighten the bounds? Double jumps?

Several examples



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 1998.

Answer #1:

Answer #2:PageRank denotes

quantitative correlation

between pairs of vertices.See slices of last year’s talk at http://math.ucsd.edu/~fan

What does a sweep of PageRank look like?

Several examples



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

strategyArcante, 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



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 ….

Documents

Fan Chung Graham University of California, San Diego