Fan Chung Graham

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New Directions in Graph Theory
for network sciences
Fan Chung Graham University of
California, San Diego
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A graph G (V,E)
edge
vertex
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Graph models

• Edges
• flights
• pairs of friends
• coauthorship
• phone calls
• linkings
• regulatory aspects

• Vertices
• cities
• people
• authors
• telephones
• web pages
• genes

_____________________________
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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?
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Graph Theory has 250 years of history.
Theory applications
Real world large graphs
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Geometric graphs
Algebraic graphs
real graphs
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Massive data
Massive graphs

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

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The Opte project
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An Internet routing (BGP) graph
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A subgraph of the Hollywood graph.
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An induced subgraph of the collaboration graph
with authors of Erdös number 2.
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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?

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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
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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

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Classical random graphs
Same expected degree for all vertices
Random graphs with specified degrees
Random power law graphs
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Some prevailing characteristics of large
realistic networks

• Sparse

• Small world phenomenon

Small diameter/average distance
Clustering

• Power law degree distribution

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3
3
4
4
edge
2
4
vertex
Degree sequence (4,4,4,3,3,2)
Degree distribution (0,0,1,2,3)
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A crucial observation

• Massive graphs satisfy the power law.

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.

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The history of the power law

• Zipfs law, 1949. (The nth most frequent word
  occurs at rate 1/n)
• Yules law, 1942.
• Lotkas 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
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Power law graphs
Power decay degree distribution.
The degree sequences satisfy a power law
The number of vertices of degree j is
proportional to j-ß where ß is some constant 1.
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Comparisons
From simulation
From real data
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The distribution of the connected components in
the Collaboration graph
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The distribution of the connected components in
the Collaboration graph
The giant component
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Examples of power law

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

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Degree distribution of an Internet graph
A power law graph with ß 2.2
Faloutsos et al 99
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Degree distribution of Call Graphs
A power law graph with ß 2.1
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The collaboration graph is a power law graph,
based on data from Math Reviews with 337451
authors
A power law graph with ß 2.25
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• 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?)
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What is the shape of a network ?
experimental
modeling
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Massive Graphs
Random graphs
Similarities Adding one (random) edge at a
time.
Differences
Random graphs almost regular.
Massive graphs uneven degrees,
correlations.
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Random Graph Theory
How does a random graph behave?
Graph Ramsey Theory
What are the unavoidable patterns?
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A random graph G(n,p)

• G has n vertices.

• For any two vertices u and v in G, au,v is
  an edge with probability p.

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What does a random graph look like?
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Prob(G is connected)?
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Prob(G is connected)
no. of connected graphs
total no. of graphs
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A random graph has property P
Prob(G has property P)
as
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Random graphs with expected degrees wi
wi expected degree at vi
Prob( i j) wiwj p
Choose p 1/?wi , assuming max wi2lt ?wi .
Erdos-Rényi model G(n,p) The special
case with same wi for all i.
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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.
/
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Barabasi 1999
Broder 2000
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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.
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Exponents for Large NetworksP(k)k -?
Networks WWW Actors Citation Index Power Grid Phone calls
? 2.1 (in) 2.5 (out) 2.3 3 4 2.1
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Properties of
ChungLu PNAS02
Random power law graphs
? gt 3 average distance
diameter c log n
log n /
log
? 3 average distance log n / log
log n diameter c
log n
2 lt ? lt 3 average distance log log n
diameter c log n
provided d gt 1 and max deg large
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The structure of random power law graphs
2 lt ? lt 3
Octopus
Core has width log log n
core
legs of length
log n
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Yahoo IM graph
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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

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Motivation
2008
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Motivation
Random spanning trees have large diameters.
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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
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The spectrum of a graph
Adjacency matrix
Many ways to define the spectrum of a graph

How are the eigenvalues related to
properties of graphs?
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The spectrum of a graph

• Adjacency matrix

• Combinatorial Laplacian

adjacency matrix

diagonal degree matrix

• Normalized Laplacian

Random walks Rate of convergence
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The spectrum of a graph
Discrete Laplace operator ? on f V ? R
For a path
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The spectrum of a graph
Discrete Laplace operator ? on f V ? R
not symmetric in general

• Normalized Laplacian

symmetricnormalized
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Properties of Laplacian eigenvalues of a graph
Spectral bound ?
holds iff G is disconnceted or bipartite.
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Question
What is the diameter of a random spanning tree of
a given graph G ?
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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

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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
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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

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A disease contact graph
Jim Walker 2008
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For a given graph G,
Gp
Contact graph
retain each edge with probability p.
infection rate
Percolation on G a random subgraph of G.
Example GKn, 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?
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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
Bollobás et. al. 2008

• dense graphs

• complete graphs

Erdös-Rényi 1959
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Percolation on special graphs or dense graphs
Percolation on general sparse graphs
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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.
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Percolation on general sparse graphs
Theorem (ChungHorn 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?
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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

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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 years talk at
http//math.ucsd.edu/fan
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What does a sweep of PageRank look like?
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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

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Michael Kearns experiments on coloring games
2006
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Michael Kearns experiments on coloring games
2006
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Classical graph coloring Chromatic graph theory
Coloring graphs in a greedy and selfish way
Coloring games on graphs
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Applications of graph coloring games

• dynamics of social networks

• conflict resolution

• Internet economics

• on-line optimization scheduling

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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?
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A graph coloring game
? the maximum degree of G
Theorem (Chaudhuri,Chung,Jamall 2008)
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.
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Improving existing methods

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

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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 .