fan chung graham university of california, san diego
Post on 19-Dec-2015
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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?
Massive dataMassive graphs
• WWW-graphs
• Call graphs
• Acquaintance graphs
• Graphs from any data a.base
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.
Examples of power law
•Inter
• Internet graphs.
• Call graphs.
• Collaboration graphs.
• Acquaintance graphs.
• Language usage
• Transportation networks
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?)
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.
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
Several examples
• Diameter of random trees of a given graph
• 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
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
The spectrum of a graph
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+∂ ∂
− −∂ ∂
The spectrum of a graph
The spectrum of a graph
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
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
• Diameter of random trees of a given graph
• Random graphs with any given degrees Diameter of random power law
graphs
• Percolation and giant components in a graph• Correlation between vertices xxxxxxxxxxxThe pagerank of a graph• Graph coloring and network games
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
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.
Percolation on general sparse graphs
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
• Diameter of random trees of a given graph
• Random graphs with any given degrees Diameter of random power law
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
Several examples
• Diameter of random trees of a given graph
• Random graphs with any given degrees Diameter of random power law
graphs
• Percolation and giant components in a graph
• Graph coloring and network games
• Correlation between vertices xxxxxxxxxxxxxThe pagerank of a graphs
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
• Diameter of random trees of a given graph
• 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 • Many new directions and tools ….