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Evolution of Networks

Notes from Lectures of J.Mendes CNR, Pisa, Italy, December 2007

Eva JahoAdvanced Networking Research Group

National and Kapodistrian University of AthensFebruary 2008

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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

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Presentation Outline (cntd.)

Non-equilibrium random networks (cntd.) Barabasi-Albert model Linear preference - general case How does the general case relates to the Barabasi-

Albert 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

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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)

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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)

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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

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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:

The average degree of a randomly chosen node in a network of size N is:

N

s

NskpN

NkP1

),,(1

),(

k

NkkPk ),(

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Degree distribution (cntd.)!/)( kkekP

kk

kekP )(

a) Poisson: , e.g. a classical random equilibrium graph of Erdos and Renyi when the total number of vertices is infinite

b) Exponential: , e.g. a citation graph with attachment of new vertices to randomly chosen old ones

c) Power-law: , e.g. a citation graph with attachment of new vertices to preferentially chosen old ones

kkP )(

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Clustering

The clustering coefficient characterizes the “connectedness” of the environment close to a vertex.

ni: number of connections among the neighbors

ki(ki-1)/2: number of possible connections among the

neighbors.

2)1(

ii

ii kk

nC

i

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Clustering (cntd.)

The average value reflects how connected are the neighboring nodes

also shows the “density” of small loops of length 3

of a tree is 0, of a fully connected graph (clique) is 1

C

C

i

iCNC

1

C

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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

llPl )(

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Average length (cntd.)

In a tree-like network about vertices are at distance l or closer from a vertex. The average length is:

Tree-like networks have the small-world effect: average length l is still small for large size of networks N

lk

k

NlkN

l

ln

ln

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k-core

k-core: maximum sub-graph of a graph whose vertices have degree at least k within this sub-graph

Cycle: 2-coreTree: 1-core Clique of 4 nodes: 3-core

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k-core (cntd.)

Pruning rule:1) Remove from the graph all vertices with degree

smaller than k2) Some vertices may have now less than k degree3) Prune again the vertices 4) Repeat until no more pruning is possible

Example of 2-core:

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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

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Small-world networks

Small-world networks combine high clustering of regular lattices “small-world effect” (small average shortest-

path length) of random networks Construction

From regular lattices by rewiring bonds (Watts and Strogatz (1998))

by making shortcuts between randomly chosen vertices

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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

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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

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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

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Non-equilibrium random networks (cntd.)

An example:(a)

(b)

(c)

(d)

(e)

The “oldest” vertices are the most connected and degrees of new nodes are the smallest

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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)

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Growing exponential networks

Initial network s: vertex, t: time step

At each time step t A new vertex is attached to a randomly

chosen vertex There are t vertices and t edges

New vertex

Old net

Randomly chosen vertex

s=1

s=2

t=2

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Growing exponential networks (cntd.)

p(k,s,t): probability that vertex s has degree k at time t

Evolution:

Initial condition:

Boundary condition:

1 1( , , 1) ( 1, , ) 1 ( , , )p k s t p k s t p k s t

t t

,2( , 1, 2, 2) kp k s t

,1( , , 2) kp k s t t

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Growing exponential networks (cntd.)

P(k,t): probability that a randomly chosen vertex has degree k at time t

Initial condition:

: stationary degree distribution Stationary equation:

( ) ( , )P k P k t

1

1( , ) ( , , )

t

s

P k t p k s tt

,2( , 2) kP k t

,12 ( ) ( 1) kP k P k ( ) 2 kP k

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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

New vertex

Old net

Preferentially chosen vertex

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The Barabasi-Albert model (cntd.)

k/(2t): probability that vertex s of degree k gets a new connection at time t (2t, total degree)

Initial condition:

Boundary condition: Stationary equation:

1( , , 1) ( 1, , ) 1 ( , , )

2 2

k kp k s t p k s t p k s t

t t

,2( , 1, 2, 2) kp k s t

,1( , , 2) kp k s t t

,1

1( ) [ ( ) ( 1) ( 1)]

2 kP k kP k k P k ( )P k k (γ=3)

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Linear preference - general case

Directed growing network, considering only in-degree 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, positive constant)

New vertex

m

Old net

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Linear preference – general case (cntd.)

Equation for the average in-degree of individual vertices:

where

Solution:

Power-law:

0

( , ) ( , )

[ ( , ) ]t

q s t q s t Am

tdu q u t A

0

[ ( , ) ]t

du q u t A mt At

( , ) , (0 1) s m

q s t wheret m A

(1 1/ )( )

2 / ( 2) ( 1) 1

P q q q

A m and

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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

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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 Barabasi-Albert model

Old net

Preferentially chosen vertices

New vertex

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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

New vertex

n

Old net

m nr

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Mixture of random and preferential attachments (cntd.) Equation for the average in-degree of individual

vertices:

where

The degree distribution of the network will remain power-law β=m/(m+n+nr+A) γ=2+(n+nr+A)/m

0

( , ) 1 ( , )

[ ( , ) ]r t

q s t q s t An m

t tdu q u t A

0

[ ( , ) ] ( )t

rdu q u t A m n n t At

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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

WWWNew Web document

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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

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Accelerated growth of networks

Accelerated growth ≡ average degree grows Assume (growth exponent a>0)

( )

Degree distribution is power-law with k0(t)≤k≤kcut(t), z>0:

Normalization:

To converge:

atk

kttkP z),(

1 atL

1)(0

tk

z dkkt

)1/(0 )( 1 zttkand

2

NkL

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Accelerated growth of networks (cntd.)

Cut-off position:

for any γ>1 ->

1) 1<γ<2

1)(

tk

z

cut

dkktt

)1/()1()( zcut ttk

)1/()1(1)1/()1(

zt za tdkkkttz

1)1/()1( az 21

2

)1()()( t

tttLttk a

cut

1

11

a

z

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Accelerated growth of networks (cntd.) For γ<2, z<a<1

If z -> 0 Only kcut increases with time, P(k) stationary

2) γ>2

• For γ>2• z>a>0• P(k) not stationary

1

11

a

)1/()2()1/(

zz

t

za tdkkkttz

)1(

)2(

zzaa

z1

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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

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Example of an accelerated growth (cntd.)

For the total number of edges L(t):

If p<1/2 Stationary degree distribution

If p>1/2 Non-stationary degree distribution

)(1)()1( tkptLtL

)(1)]([2

1/)(2)( tkptkt

dt

dttLtk

)21/(2 pk 3/11 p

12)( pttk 3)1/(11 p 2)1/(1 pz

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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

New word

Old word web

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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 scale-free networks

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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