Transcript Lectures of Prof. Mentes

Evolution of Networks
Notes from Lectures of J.Mendes
CNR, Pisa, Italy, December 2007
Eva Jaho
Advanced Networking Research Group
National and Kapodistrian University of Athens
February 2008
1
Presentation Outline
 Basic graph notions






Tree
Loops
Degree distribution
Clustering
Average length
k-core
 Equilibrium random networks
 Small-world networks
 Watts-Strogatz model
 Non-equilibrium random networks
 A citation graph
 Growing exponential networks
2
Presentation Outline (cntd.)
 Non-equilibrium random networks (cntd.)
 Barabasi-Albert model
 Linear preference - general case
 How does the general case relates to the BarabasiAlbert model?
 Mixture of random and preferential attachments
 Growth of the WWW
 Accelerated growth of networks
 Example of an accelerated growth
 Evolution of language
 Failures and attacks
3
Tree
 A tree is a particular kind of graphs
without loops. A connected tree is a tree
with no separate parts.
 In a tree: N = L+1 (N vertices, L edges)
4
Loops
 In a connected undirected graph, the
number I of loops is related to N and L.
 I = L+1-N
 Two loops in the figure (cannot be
reduced to smaller ones)
5
Degree distribution
 In a directed network (directed edges),
 in-degree ki: number of incoming edges of a
vertex
 out-degree ko: number of outgoing edges of
a vertex
 k = ki+ko: degree, the total number of
connections.
ki
ko
6
Degree distribution (cntd.)
 p(k,s,N): probability that the vertex s in the
network of size N has k connections (k nearest
neighbours)
 The degree distribution of a randomly chosen
node is:
N
P( k , N ) 
1
p ( k , s, N )

N s 1
 The average degree of a randomly chosen
node in a network of size N is:
k   kP(k , N )
k
7
Degree distribution (cntd.)
k
a) Poisson: P(k )  e k k / k! , e.g. a classical random equilibrium graph
of Erdos and Renyi when the total number of vertices is infinite
k
b) Exponential: P(k )  e , e.g. a citation graph with attachment of
new vertices to randomly chosen old ones
c) Power-law: P(k )  k , e.g. a citation graph with attachment of new
vertices to preferentially chosen old ones
8
Clustering
 The clustering coefficient characterizes the “connectedness”
of the environment close to a vertex.
ni
Ci 
ki (ki  1)
2
 ni: number of connections among the neighbors
 ki(ki-1)/2: number of possible connections among the
neighbors.
i
9
Clustering (cntd.)
 The average value C reflects how connected
are the neighboring nodes
1
C
N
C
i
i
 C also shows the “density” of small loops of
length 3
 C of a tree is 0, of a fully connected graph
(clique) is 1
10
Average length
 Distance between two vertices = length of the
shortest path between them
 Distances l are distributed with some
distribution function P(l)
 P(l): the probability that the length of the shortest
path between two randomly chosen vertices is l
 Average length of the shortest path
l   lP(l )
l
11
Average length (cntd.)
l
 In a tree-like network about k vertices are at
distance l or closer from a vertex. The average
length is:
ln N
N  k l 
ln k
l
 Tree-like networks have the small-world effect:
average length l is still small for large size of
networks N
12
k-core
 k-core: maximum sub-graph of a
graph whose vertices have degree at
least k within this sub-graph
Tree: 1-core
Cycle: 2-core
Clique of 4 nodes:
3-core
13
k-core (cntd.)
 Pruning rule:
1) Remove from the graph all vertices with degree
smaller than k
2) Some vertices may have now less than k degree
3) Prune again the vertices
4) Repeat until no more pruning is possible
 Example of 2-core:
14
Equilibrium random networks
 Example: The classical random graph
(the Erdos-Renyi model)
 Rules:
(a) The total number of vertices is fixed
(b) Randomly chosen vertices are connected
15
Small-world networks
 Small-world networks combine
 high clustering of regular lattices
 “small-world effect” (small average shortestpath length) of random networks
 Construction
 From regular lattices by rewiring bonds
(Watts and Strogatz (1998))
 by making shortcuts between randomly
chosen vertices
16
Watts-Strogatz model
 All the edges of a regular lattice are rewired
with a probability p to randomly chosen
vertices
 p small, similar to original regular lattice
 p large, similar to the classical random
graph
p=0
p small
p large
17
Watts-Strogatz model (cntd.)
 Normalized average shortest-path length L(p)/L(0) and
clustering coefficient C(p)/C(0) vs the fraction p of rewired
connecions
small-world
18
Non-equilibrium random networks
 A random graph growing through the
simultaneous addition of vertices and
edges
 Rules:
(a) At each time step, a new vertex is added to
the graph
(b) Simultaneously, a pair of randomly chosen
vertices (or several pairs) is connected by
an edge
19
Non-equilibrium random networks
(cntd.)
 An example:
(a)
(b)
 The “oldest” vertices are
the most connected and
degrees of new nodes are
the smallest
(c)
(d)
(e)
20
A citation graph
 At each time step, a new vertex is added to the graph
 It is connected with some old node via an edge
(a)
(b)
(c)
(d)
(e)
21
Growing exponential networks
s=2
 Initial network
 s: vertex, t: time step
 At each time step t
s=1
t=2
 A new vertex is attached to a randomly
chosen vertex
 There are t vertices and t edges
Old net
New vertex
Randomly chosen vertex
22
Growing exponential networks (cntd.)
 p(k,s,t): probability that vertex s has degree k
at time t
 Evolution:
1
 1
p(k , s, t  1)  p(k  1, s, t )  1   p(k , s, t )
t
 t
 Initial condition:
p(k , s  1, 2, t  2)  k ,2
 Boundary condition:
p(k , s  t, t  2)   k ,1
23
Growing exponential networks (cntd.)
 P(k,t): probability that a randomly chosen vertex
has degree k at time t
1 t
P ( k , t )   p ( k , s, t )
t s 1
 Initial condition:
P(k , t  2)   k ,2
 P(k )  P(k , t  ) : stationary degree distribution
 Stationary equation:
2P(k )  P(k 1)  k ,1
P( k )  2  k
24
The Barabasi-Albert model
 Barabasi and Albert (1999) combined
the growth and preferential linking
 At each time step t
 A new vertex is attached to an old one with
a probability proportional to its degree
Old net
New vertex
Preferentially
chosen vertex
25
The Barabasi-Albert model (cntd.)
 k/(2t): probability that vertex s of degree k gets a new
connection at time t (2t, total degree)
k 1
k

p(k , s, t  1) 
p(k  1, s, t )  1   p(k , s, t )
2t
 2t 
 Initial condition:
p(k , s  1, 2, t  2)  k ,2
 Boundary condition:
 Stationary equation:
p(k , s  t, t  2)   k ,1
1
P(k )  [kP(k )  (k  1) P(k  1)]   k ,1
2
P(k )  k 
(γ=3)
26
Linear preference - general case
 Directed growing network, considering only indegree distributions q=ki
 At each time step, a new vertex is added to the
network
 m: incoming links (don’t care about source ends)
 The probability that a new edge becomes attached to
some vertex of in-degree q is proportional to q+A (A,
m
positive constant)
New vertex
Old net
27
Linear preference – general case
(cntd.)
 Equation for the average in-degree of individual
vertices:
 q ( s, t )
m
t
 where
t
q ( s, t )  A
t
 du[q(u, t )  A]
0
 du[q(u, t )  A]  mt  At
0

m
s
q
(
s
,
t
)

,
where


(0    1)
 Solution:
 
m A
t
 (11/  )

P
(
q
)

q

q

 Power-law:
  2  A / m (  2) and  (  1)  1
28
Linear preference – general case
(cntd.)
 While parameter A increases from 0 to
∞
 γ increases from 2 to ∞
 β decreases from 1 to 0
 If A → 0
 Most connections will be to the oldest vertex
 If A → ∞
 Preference is absent
 Similar to exponential growing network
29
How does the general case relate to
the Barabasi-Albert model?
 At each time step a new vertex has m outgoing
connections
 Old vertices obtain only incoming links
 Degree of an arbitrary vertex: k=q+m
 In general case if A=m (β=1/2 and γ=3)

the probability of attaching a new edge - proportional to
the degree of a vertex (q+A=q+m=k), that is BarabasiAlbert model
Old net
New vertex
Preferentially
chosen vertices
30
Mixture of random and preferential
attachments
 At each time step a new vertex with n incoming links is
added
 Simultaneously, m new edges become attached to
preferentially chosen vertices
 And in addition, nr new edges become attached to
randomly chosen vertices
n
New vertex
m
nr
Old net
31
Mixture of random and preferential
attachments (cntd.)
 Equation for the average in-degree of individual
vertices:
 q ( s, t )
1
 nr  m
t
t
q( s, t )  A
t
 du[q(u, t )  A]
0
t
 where
 du[q(u, t )  A]  (m  n  n )t  At
r
0
 The degree distribution of the network will remain
power-law


β=m/(m+n+nr+A)
γ=2+(n+nr+A)/m
32
Growth of the WWW
 WWW is an array of documents (pages) connected by
hyperlinks, which are mutual references in these
documents



A new Web document must have at least one incoming
hyperlink to be accessible
It has several (or none) references to existing documents
of the WWW
Old pages can be updated, and so new hyperlinks can
appear between them
WWW
New Web
document
33
Growth of the WWW (cntd.)
 β=m/(m+n+nr+A)
 γ=2+(n+nr+A)/m
 From experimental data
 (m+n+nr)≈10
 n=1 and if A+nr=0
 γ≈2.1
34
Accelerated growth of networks
 Accelerated growth ≡ average degree grows
 Assume k  t a (growth exponent a>0)
N
a 1
 Lt
( Lk
)
2
 Degree distribution is power-law with
k0(t)≤k≤kcut(t), z>0: P(k , t )  t z k 

 Normalization:
z 
t
 k dk  1
k0 ( t )
 To converge:
  1 and k0 (t )  t z /( 1)
35
Accelerated growth of networks
(cntd.)
 Cut-off position: t

z 
t
 k dk  1
k cut ( t )
 for any γ>1 ->
kcut (t )  t ( z 1) /( 1)
1) 1<γ<2
t ( z1) /(  1)
ta  
kt z k  dk  t 1( z 1) /( 1)
( z  1) /(  1)  a  1
z 1
  1
a 1
kcut (t )  t
a 1
t (t  1) 2
 L(t ) 
t
2
36
Accelerated growth of networks
(cntd.)
 For γ<2, z<a<1

If z -> 0

Only kcut increases with time, P(k) stationary
  1
2) γ>2
1
a 1
t a   z /(  1) kt z k  dk  t z  z ( 2) /( 1)
t
(  2)
a zz
(  1)
•
  1
z
a
For γ>2
•
•
z>a>0
P(k) not stationary
37
Example of an accelerated growth
 Undirected citation graph
 A new vertex is attached to a randomly
chosen old one
 plus to some of its nearest neighbors, to
each one of them with probability p
Old net
New vertex
38
Example of an accelerated growth
(cntd.)
 For the total number of edges L(t):
L(t  1)  L(t )  1  pk (t )
1 d
k (t )  2 L(t ) / t 
[t k (t )]  1  p k (t )
2 dt
 If p<1/2

Stationary degree distribution
k  2 /(1  2 p)
  1  1/ p  3
 If p>1/2

Non-stationary degree distribution
k (t )  t 2 p1
  1  1 /(1  p)  3
z  1 /(1  p)  2
39
Evolution of language
 The total number of connections
between words grows more rapidly than
the number of words in the language
 Accelerated growth of Word Web
Old word web
New word
40
Failures and attacks
 The integrity of a network depends on the giant
(largest connected) component
 A network is damaged by eliminating its giant
component
 Simulations of the random damage (failure)
and attack on exponential and power-law
networks
 Cluster size distribution for different f (fraction of
deleted vertices)
 For small f under failure or attack
 There is still a large cluster in exponential and scalefree networks
41
Failures and attacks (cntd.)
 For large f under failure


Elimination of the giant
component in
exponential network
There is still a large
cluster in scale-free
network
 Scale-free nets under
attack behave similarly
as exponential nets
under attack or failures
 Scale-free nets are
robust to random
failures
42