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New Directions in Graph Theory. for network sciences. Fan Chung Graham University of California, San Diego. A graph G = ( V, E ). edge. vertex. Vertices cities people authors telephones web pages genes. Edges flights pairs of friends coauthorship

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slide1

New Directions in Graph Theory

for network sciences

Fan Chung Graham

University of California, San Diego

graph models
Vertices

cities

people

authors

telephones

web pages

genes

Edges

flights

pairs of friends

coauthorship

phone calls

linkings

regulatory aspects

Graph models

_____________________________

slide4

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?

slide5

Graph Theory has 250 years of history.

Theory applications

Real world large graphs

slide6

Geometric graphs

Algebraic graphs

real graphs

slide7

Massive data

Massive graphs

  • WWW-graphs
  • Call graphs
  • Acquaintance graphs
  • Graphs from any data a.base
slide12

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?
slide13

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

slide14

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
slide15

Classical random graphs

Same expected degree for all vertices

Random graphs with specified degrees

Random power law graphs

slide16

Some prevailing characteristics of large realistic networks

  • Sparse
  • Small world phenomenon

Small diameter/average distance

Clustering

  • Power law degree distribution
slide17

3

3

4

4

edge

2

4

vertex

Degree sequence: (4,4,4,3,3,2)

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

a crucial observation

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.

the history of the power law

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

slide20

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.

slide21

Comparisons

From real data

From simulation

the distribution of the connected components in the collaboration graph23
The distribution of the connected componentsin the Collaboration graph

The giant component

examples of power law
Examples of power law
  • Inter
  • Internet graphs.
  • Call graphs.
  • Collaboration graphs.
  • Acquaintance graphs.
  • Language usage
  • Transportation networks
slide25

Degree distribution of an Internet graph

A power law graph with β = 2.2

Faloutsos et al ‘99

slide26

Degree distribution of Call Graphs

A power law graph with β = 2.1

slide27

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

slide28

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?)

slide30

Massive Graphs

Random graphs

Similarities: Adding one (random) edge at a time.

Differences:

Random graphs almost regular.

Massive graphs uneven degrees,

correlations.

slide31

Random Graph Theory

How does a random graph behave?

Graph Ramsey Theory

What are the unavoidable patterns?

slide32

Paul ErdÖs and A. Rényi,

On the evolution of random graphs

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

slide33

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.
a random graph has property p
A random graph has property P

Prob(G has property P)

as

slide38

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.

slide39

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

slide40

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.

slide42

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’

slide43

The structure of random power law graphs

2 << 3

`Octopus’

Core has width log log n

core

legs of length

log n

slide45

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
slide47

Motivation

Random spanning trees have large diameters.

slide48

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

slide49

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?

slide50

The spectrum of a graph

  • Adjacency matrix
  • Combinatorial Laplacian

adjacency matrix

diagonal degree matrix

  • Normalized Laplacian

Random walks

Rate of convergence

slide51

The spectrum of a graph

Discrete Laplace operator ∆ on f: V R

For a path

slide52

{

{

The spectrum of a graph

Discrete Laplace operator ∆ on f: V R

not symmetric in general

  • Normalized Laplacian

symmetricnormalized

properties of laplacian eigenvalues of a graph

Properties of Laplacian eigenvalues of a graph

Spectral bound  :

“=“ holds iff Gis disconnceted or bipartite.

question

Question

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

some notation

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,
  •  : the minimum degree,
  • d =vol(G)/n: the average degree,
  • The second-order average degree
diameter of random spanning trees

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

If

then

slide57

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 xxxxxxxxxxxThe pagerank of a graph
  • Graph coloring and network games
slide59

Gp :

For a given graph G,

Contact graph

retain each edge with probability p.

infection rate

Percolation on G = a random subgraph of G.

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

Question:For what p, does Gphave a giant xxxxxxxxxcomponent?

Under what conditions will the disease spread to a large population?

slide60

Percolation on graphs

History: Percolation on

  • lattices

Hammersley 1957, Fisher 1964 ……

  • hypercubes

Ajtai, Komlos, Szemerédi 1982

  • Cayley graphs

Malon, Pak 2002

  • d-regular expander graphs

Frieze et. al. 2004

Alon et. al. 2004

  • dense graphs

Bollobás et. al. 2008

  • complete graphs

Erdös-Rényi 1959

slide61

Percolation on special graphs or dense graphs

Percolation on general sparse graphs

slide62

Percolation on general sparse graphs

Theorem(Chung,Horn,Lu 2008)

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

provided that the maximum degree of ∆satisfies

under some mild conditions.

slide63

Percolation on general sparse graphs

Theorem(Chung+Horn +Lu)

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

provided that the maximum degree of ∆satisfies

under some mild conditions.

Questions: Tighten the bounds? Double jumps?

slide64

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 xxxxxxxxxxxxxThe pagerank of a graphs
  • Graph coloring and network games
slide66

What is PageRank?

Answer #1:

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 #2:

PageRank denotes quantitative correlation between pairs of vertices.

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

slide68

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 xxxxxxxxxxxxxThe pagerank of a graphs
  • Graph coloring and network games
slide71

Classical graph coloring

Chromatic graph theory

Coloring graphs in a greedy and selfish way

Coloring games on graphs

slide72

Applications of graph coloring games

  • dynamics of social networks
  • conflict resolution
  • Internet economics
  • on-line optimization + scheduling
slide73

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?

slide74

A graph coloring game

∆ : the maximum degree of G

Theorem(Chaudhuri,Chung,Jamall 2008)

If ∆+2colors are available, the coloring game converges in O(log n) rounds.

If ∆+1colors are available, the coloring game may not converge for some initial settings.

improving existing methods

Improving existing methods

  • Probabilistic methods, random graphs.
  • Random walks and the convergence rate
  • Lower bound techniques
  • General Martingale methods
  • Geometric methods
  • Spectral methods
slide76

New directions in graph theory

  • 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
  • Many new directions and tools ….
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