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## PowerPoint Slideshow about 'Fan Chung Graham University of California, San Diego' - zanthe

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### A crucial observation

### The history of the power law

### Properties of Laplacian eigenvalues of a graph

### Question

### Some notation

### Diameter of random spanning trees

### Improving existing methods

New Directions in Graph Theory

for network sciences

Fan Chung Graham

University of California, San Diego

Vertices

cities

people

authors

telephones

web pages

genes

Edges

flights

pairs of friends

coauthorship

phone calls

linkings

regulatory aspects

Graph models_____________________________

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?

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

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?

- 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

- 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

Same expected degree for all vertices

Random graphs with specified degrees

Random power law graphs

Some prevailing characteristics of large realistic networks

- Sparse

- Small world phenomenon

Small diameter/average distance

Clustering

- Power law degree distribution

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.

- 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 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.

The distribution of the connected componentsin the Collaboration graph

The distribution of the connected componentsin the Collaboration graph

The giant component

Examples of power law

- Inter
- Internet graphs.
- Call graphs.
- Collaboration graphs.
- Acquaintance graphs.
- Language usage
- Transportation networks

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 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?)

Random graphs

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

Differences:

Random graphs almost regular.

Massive graphs uneven degrees,

correlations.

How does a random graph behave?

Graph Ramsey Theory

What are the unavoidable patterns?

On the evolution of random graphs

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

- G has n vertices.

- For any two vertices u and v in G, a{u,v} is anedge with probability p.

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.

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

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.

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’

The structure of random power law graphs

2 << 3

`Octopus’

Core has width log log n

core

legs of length

log n

- 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

2008

Random spanning trees have large diameters.

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

Many ways

to define

the spectrum of a graph

Adjacency matrix

How are the eigenvalues related to

properties of graphs?

- Adjacency matrix

- Combinatorial Laplacian

adjacency matrix

diagonal degree matrix

- Normalized Laplacian

Random walks

Rate of convergence

{

The spectrum of a graph

Discrete Laplace operator ∆ on f: V R

not symmetric in general

- Normalized Laplacian

symmetricnormalized

Spectral bound :

“=“ holds iff Gis disconnceted or bipartite.

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

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

Chung, Horn and Lu 2008

If

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

If

then

- 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

Jim Walker 2008

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?

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

Percolation on special graphs or dense graphs

Percolation on general sparse graphs

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.

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?

- 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

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

- 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

Chromatic graph theory

Coloring graphs in a greedy and selfish way

Coloring games on graphs

Applications of graph coloring games

- dynamics of social networks

- conflict resolution

- Internet economics

- on-line optimization + scheduling

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?

∆ : 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.

- 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

- 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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