W eierstraß -In stitu t fü r A n g ew an d te A n alysis u n d Stoch astik
WIAS Kolloquium
X. YAO
Random Graph and Complex Networks
Mohrenstr 39 10117 Berlin
[email protected]
www.wias-berlin.de
June 26, 2006
The Historical Overview
. Erdös-Rényi Models for Random Graph
– Probabilistic methods
– phase transition and critical phenomena in random graphs
. Small World Phenomena
– six-degree separation of the “world”.(S. Milgram 1967 & F. Karinthy 1929)
– Watts-Strogatz model
– The case of Internet: 19 degrees.(Albert et al. 1999)
. Scale-free Networks
– The power-law degree distribution of the real world networks
– Barabasi-Albert Model, the generalized models...
. Complex networks
WIAS Kolloquium
June 26, 2006
2(30)
Birth of the random graph
. Erdös-Rènyi model
Vertex set: V = {1, 2, · · · , n};
– The binomial random graph Gn,p
◦ The graph space Ω = { the graphs with the vertex set V };
◦ The σ-algebra on Ω: F = { the subsets of Ω };
◦ The probability measure P
n
(
P(G) = p (1 − p) 2)−eG , G ∈ Ω.
eG
(1)
– The uniform random graph Gn,M
◦ The graph space Ω = { the graphs with the vertex set V and with M edges };
◦ The σ-algebra on Ω: F = { the subsets of Ω };
◦ The probability measure P
µ¡n¢¶−1
2
P(G) =
, G∈Ω
(2)
M
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June 26, 2006
3(30)
The Problems of Random Graphs
. Subgraphs problem
– The existence in the Gn,p(or Gn,M ) of at least one copy of a given graph G.
. Mathchings
– matching: A matching on a graph is a set of edges such that no two of them
share a vertex in common
– perfect matching: the matching covers every vertex of the graph.
– The existence of perfect matchings in random graph
. The Phase transition and the critical behavior
– connectivity problem, largest component size, ...
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June 26, 2006
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Some important properties
. Degree distributions
E(Xk)/N
Xk/N
0.10
0.05
<k>
0.00
0
10
20
30
k
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June 26, 2006
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Some important properties
. Clustering degree ∆ and Clustering Coefficients C
∆v = the number of links between
the neighbors of vertex v,
¡k v ¢
Cv = ∆v / 2 ,
C = Cv /
WIAS Kolloquium
June 26, 2006
P
v∈V
6(30)
Cv
Some important properties
. Mean shortest path `
D15, 1 → 2 → 5,
D17, 1 → 2 → 5 → 6 → 7.
To describe the “diameter” of the
network, we use the mean value
of the shortest paths between any
two vertices.
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June 26, 2006
7(30)
The evolution of the random graphs
When the connecting probability p increases from 0 to 1, the
corresponding random graph become denser and denser. For
many graph property, there exists
a threshold to decide the whether
the random graph possess such
property.
p=0
p=0.1
WIAS Kolloquium
p=0.15
June 26, 2006
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Probabilistic methods
In order to prove the existence of an object with a specific property, prove that a
randomly chosen object satisfies that property with positive probability.
The example: the connectivity problem (Erdös and Rényi(1959))
.– For the binomial random graphs Gn,p, where p =
log n+c+o(1)
n
−c
P(Gn,pis connected) → e−e
and we have the critical point p∗ =
log n
n
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June 26, 2006
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where c ∈ R, we have
(3)
A generalized case: Random Intersection graphs
. Example 1: metabolic networks
A chemical reaction need several kinds of molecules, and one kind of molecule
may involved with many chemical reactions;
(Barabási A.(2005))
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June 26, 2006
10(30)
A generalized case: Random Intersection graphs
. Example 2: Scientific Collaboration Networks
An scientific article may have several authors, and an author may cooperated
with many others in other publications.
If the authors are looked as vertices and connected two authors if they have
published some same articles together, we also induce a graph.
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June 26, 2006
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A generalized case: Random Intersection graphs
. The model of random intersection graph GN,M,p. (Siger-Cohen, K., 1995)
V = {v1, v2, · · · , vN }, W = {w1, w2, · · · , wM }, pvw ∈ [0, 1].
The relations among the parameters M, N, p decide almost all the
properties of the GN,M,p, let.
M = N α, p = N −β
where
α ≥ 0 and β ≥ 0
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June 26, 2006
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A generalized case: Random Intersection graphs
. Some results of random intersection graphs
– subgraph problem(Singer-Cohen, K.(1995,1999), Fill(2000))
– Degree distribution(Stark, D. (2005))
– Clustering Coefficients(Yao.X, etc. (2006))
WIAS Kolloquium
June 26, 2006
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The critical behavior of the clustering property in GN,M,p
. The localized clustering property (1): clustering degree ∆v
– Lv : the number of links between the vertex
v and W set.
5
7
x 10
6
– ∆v : the number of links between the
neighbors of vertex v in GN,M,p.
4
6000
E( |∆ | | L )
3
v
E( |∆v| | L )
5
2
4000
2000
1
0
0
0
−1
0
10
20
30
L
50
100
150
L
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200
250
 2
c L 2−2β



N
,
β ≤ α < 2β


h ¯ i  2
¯
(1 − e−Lp)3 2
E ∆v ¯L ∼
N , α = 2β

2


2


 N ,
α > 2β
2
(4)
June 26, 2006
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The critical behavior of the clustering property in GN,M,p
. The localized clustering property (2): Clustering Coefficients Cv .
– Lv : the number of links between the vertex
v and W set.
1.1
1
– Cv : the Clustering Coefficient of vertex v.
E( Cv | L )
0.9
0.8

1


β ≤ α < 2β
h ¯ i  L,
¯
E Cv ¯L ∼
− cLβ
1 − e N , α = 2β



1,
α > 2β
0.7
0.6
0.5
0.4
0.3
(5)
0.2
0.1
0
50
100
150
200
250
L
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June 26, 2006
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The critical behavior of the clustering property in GN,M,p
. The Global clustering property: Clustering Coefficients C.
– Lv : the number of links between the vertex
v and W set.
1
0.9
0.8
– Cv : the Clustering Coefficient of vertex v.
0.7
ECv
0.6


c0,


 1 −(α−β)
h i 
N
,
E Cv ∼
c

−c2

1
−
e
,


 1,
0.5
0.4
0.3
0.2
0.1
0
100
200
300
400
500
600
700
800
900
N
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June 26, 2006
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α=β
β < α < 2β
α = 2β
α > 2β
(6)
Small world phenomena
. The six degree separation of the world.
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June 26, 2006
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Small world phenomena
. Watts-Strogatz model for small world phenomena
a)
b)
(Watts & Strogatz, 1998)
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June 26, 2006
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Scale-free Networks
. Example 1: Citation Networks
(Barabási(1999))
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June 26, 2006
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Scale-free Networks
. Example 2: Internet
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June 26, 2006
20(30)
The real world network: Scale-free Networks
. Example 3: Broadcast board System(BBS): the formation of the networks
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June 26, 2006
21(30)
Scale-free Networks
. Example 3: Broadcast board System(BBS): the degree distribution, (Yao,X. etc.
(2005))
6
10
4
10
2
10
0
10
−2
10
−4
10
0
10
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10
2
10
June 26, 2006
3
10
22(30)
4
10
Scale-free Networks
. The Barabasi-Albert model — linear preferential attachment
– Growth: The network increases with time, by adding a new vertex at each step
with m > 0 new edges;
– Preferential attachments: The new added vertices prefer to link to highly connected vertices;
ki
P(vt connected with vi) = P
j kj
WIAS Kolloquium
June 26, 2006
23(30)
(7)
Scale-free Networks
. The Barabasi-Albert model — linear preferential attachment
– P(k) ∼ k −3, (Albert & Barabási (1999))
0
0
10
10
(b)
-3
P(k)/2m
2
10
-2
10
-5
10
-2
-7
10
10
-9
10
0
P(k)
10
1
2
10
10
k
3
10
-4
10
3
10
2
10
-4
ki(t)
10
-6
10
1
10
(a)
0
10
-6
10
-8
0
10
1
10
2
10
3
10
10
0
10
k
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10
4
t
10
1
2
10
10
k
June 26, 2006
24(30)
3
10
Scale-free Networks
. The Barabasi-Albert model — The generalized case
– Growth: The network increases with time, by adding a new vertex with m > 0
new edges;
– Preferential attachments: The new added vertices prefer to link to highly connected vertices, but with an offset k ∗;
ki + k ∗
P(vt connected with vi) = P
∗
j (kj + k )
WIAS Kolloquium
June 26, 2006
25(30)
(8)
Scale-free Networks
. The Barabasi-Albert model — The generalized case
P (k) ∼ k
WIAS Kolloquium
June 26, 2006
∗
3+ km
26(30)
Scale-free Networks
. Is the “preferential attachment” true in the real world?
(Albert & Barabasi, (2001))
WIAS Kolloquium
June 26, 2006
27(30)
What is complex networks
. Power-law degree distribution;
. High Clustering property;
. Short diameter.
D ∼ log N
WIAS Kolloquium
June 26, 2006
28(30)
The clustering property of Barabási-Albert model(The generalized case)
. The clustering degree of Generalzed barabasi model(Yao,X. etc. (2005))
4
10
3
−3
10
10
−2
10
2
10
1
10
−3
0
10
10
−4
1
10
κ(k) ∼
C1
1/β
k
,
1−β
N
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10
where β =
3
10
1
10
2
10
3
10
10
1
10
2
10
m
2m+k ∗ .
June 26, 2006
29(30)
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The current hot points and future problems
. Constructing more general and rigorous models;
. The motif problem—The subgraph problems in complex networks;
. The community detection—classification problems.
. The dynamical model on the complex networks, e.g., random walks
WIAS Kolloquium
June 26, 2006
30(30)