Fan Chung Graham
University of California, San Diego
A graph G = (V,E)
edge
vertex
Graph models
Vertices
Edges
_____________________________
cities
people
authors
telephones
web pages
genes
flights
pairs of friends
coauthorship
phone calls
linkings
regulatory aspects
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?
Graph Theory has 250 years of history.
Real world large graphs
Theory
applications
Geometric graphs
Algebraic graphs
real graphs
Massive data
• WWW-graphs
• Call graphs
• Acquaintance graphs
• Graphs from any data
a.base
Massive graphs
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
• 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?
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
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
Classical random graphs
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
3
3
4
4
edge
2
4
vertex
5
Degree sequence: (4,4,4,3,3,2)
4
3
Degree distribution: (0,0,1,2,3)
2
1
0
degree_0 degree_1 degree_2 degree_3 degree_4
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.
The history of the power law
• Zipf’s law, 1949.
(The nth most frequent word occurs at rate 1/n)
• Yule’s law, 1942.
(City populations follow a power law.)
• Lotka’s law, 1926.
(Distribution of authors in chemical abstracts)
• Pareto, 1897
(Wealth distribution follows a power law.)
1907-1916
Natural language
Bibliometrics
Social sciences
Nature
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.
Comparisons
From real data
From simulation
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
Degree distribution of an Internet graph
A power law graph with β = 2.2
Faloutsos et al ‘99
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
(Guess who?)
•The giant component of size 208,000
•84,000 isolated vertices
What is the `shape’ of a network ?
experimental
modeling
Massive Graphs
Random graphs
Similarities: Adding one (random) edge at a time.
Differences: Random graphs
Massive graphs
almost regular.
uneven degrees,
correlations.
Random Graph Theory
How does a random graph
behave?
Graph Ramsey Theory
What are the 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
4 1
 
8 2
1
p
2
A random graph has property P
Prob(G has property P)  1
as n  .
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.
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
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.
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
Properties of
Random power law graphs
 >3
average distance
diameter
 =3
2<  <3
average distance
Chung+Lu
PNAS’02
log n / log d%
c log n
log n / log log n
diameter
c log n
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
Core has width
log log n
legs of length log n
Yahoo IM graph
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
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 n .
Theorem (Aldous 1990) :
The diameter diam(T) of a random spanning
tree in a regular graph with spectral bound 
is
c(1  ) n
c n log n
 E(diam(T )) 
.
log n
1 
The spectrum of a graph
Many ways
Adjacency matrix
to define
the spectrum of a graph
How are the eigenvalues related to
properties of graphs?
The spectrum of a graph
•Adjacency matrix
•Combinatorial Laplacian
L  D A
adjacency matrix
diagonal degree matrix
•Normalized Laplacian
Random walks
Rate of convergence
The spectrum of a graph
Discrete Laplace operator ∆ on f: V  R
1
f ( x) 
dx
For a path
 ( f ( x)  f ( y))
y x
1
f (x)   {( f (x j 1 )  f (x j ))
2
( f (x j )  f (x j 1 ))}
2
 2 f ( x)
x


{ f ( x j 1 ) 
f ( x j )}
x
x
The spectrum of a graph
Discrete Laplace operator ∆ on f: V  R
1
f ( x) 
dx
 ( f ( x)  f ( y))
y x
1
L ( x, y ) 
not symmetric in general
{ 1
•Normalized Laplacian
symmetric
normalized
L( x, y ) 
if x  y
if x  y
dx
1
{
if x  y
1
dxd y
if x  y
Properties of Laplacian eigenvalues of a graph
0  0  1      n1  2
Spectral bound  :
  max | i  1|
i0
  1 “=“ 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,
• 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
d%
2
d
x x

x
dx
Diameter of random spanning trees
Chung, Horn and Lu 2008
2
log
n
If d 
,
2
log 
then with probability 1-, a random tree T
in G has diameter diam(T) satisfying
(1  )
 nd
c
nd
 diam(T ) 
log n.
%
  log(1 /  )
d
If d% Cd, then
( n)  E(diam(T ))  O( n log n).
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
A disease contact graph
Jim Walker 2008
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 Gp have a giant
xxxxxxxxxcomponent?
Under what conditions will the disease
spread to a large population?
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
• dense graphs
Frieze et. al. 2004
Alon et. al. 2004
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
1
p
d%
provided that the maximum degree of
∆ satisfies
 d%
  o 

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
max(2d log n, ( vol G )
provided that the maximum degree of
∆ satisfies
1
 d%
under some mild conditions.
p , o
%


d

Questions: Tighten the bounds? Double jumps?
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
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
What does a sweep of PageRank look like?
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
Michael Kearns’ experiments on coloring games
2006
Michael Kearns’ experiments on coloring games
2006
Classical graph coloring
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
•
•
•
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
∆ : 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.
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
• 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 ….