mathematical models of networks - leonid zhukov · lecture outline 1 random graphs erdos-renyi...

Mathematical models of networks

Leonid E. Zhukov

School of Data Analysis and Artificial IntelligenceDepartment of Computer Science

National Research University Higher School of Economics

Social Network AnalysisMAGoLEGO course

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Lecture outline

1 Random graphsErdos-Renyi model

2 Preferential attachmentBarabasi-Albert model

3 Small world modelWatts-Strogatz model

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Network models

Empirical network features:

Power-law (heavy-tailed) degree distribution

Small average distance (graph diameter)

Large clustering coefficient (transitivity)

Giant connected component

Generative models:

Random graph model (Erdos & Renyi, 1959)

Preferential attachement model (Barabasi & Albert, 1999)

Small world model (Watts & Strogatz, 1998)

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Random graph model

Graph G{E ,V }, nodes n = |V |, edges m = |E |Erdos and Renyi, 1959.

Gn,p - each pair out of N = n(n−1)2 is connected with probability p,

number of edges m - random number

〈m〉 = pn(n − 1)

2

〈k〉 =1

n

∑i

ki =2〈m〉n

= p (n − 1) ≈ pn

ρ =〈m〉

n(n − 1)/2= p

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Node degree distribution

Poisson Distribution

P(ki = k) =λke−λ

k!, λ = pn = 〈k〉

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Random graph

〈k〉 = pn > 1

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Phase transition

Consider Gn,p as a function of p

p = 0, empty graph

p = 1, complete (full) graph

There are exist critical pc , structural changes from p < pc to p > pc

Gigantic connected component appears at p > pc

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Random graph model

p < pc p = pc p > pc

igraph:erdos.renyi.game()

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Phase transition

Graph G (n, p), for n→∞, critical value pc = 1/n

when p < pc , (〈k〉 < 1) there is no components with more thanO(ln n) nodes, largest component is a tree

when p = pc , (〈k〉 = 1) the largest component has O(n2/3) nodes

when p > pc , (〈k〉 > 1) gigantic component has all O(n) nodes

Critical value: 〈k〉 = pcn = 1- on average one neighbor for a node

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Connected component

〈k〉 = pn

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Clustering coefficient

Clustering coefficient

C (k) =#of links between NN

#max number of links NN=

pk(k − 1)/2

k(k − 1)/2= p

C = p =〈k〉n

when n→∞, C → 0

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Graph diameter

G (n, p) is locally tree-like (GCC) (no loops; low clustering coefficient)

on average, the number of nodes d steps away from a node 〈k〉d

in GCC, around pc , 〈k〉d ∼ n,

d ∼ ln n

ln〈k〉

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Random graph model

Node degree distribution function - Poisson:

P(k) =λke−λ

k!, λ = pn = 〈k〉

Average path length:

〈L〉 ∼ log(N)/ log〈k〉

Clustering coefficient:

C =〈k〉n

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Preferential attachment model

Barabasi and Albert, 1999Dynamical growth model

t = 0, n0 nodes

growth: on every step add a node with m0 edges (m0 ≤ n0),ki (i) = m0

Preferential attachment: probability of linking to existing node isproportional to the node degree ki

Π(ki ) =ki∑i ki

=ki

2m0t

after t steps: n0 + t nodes, m0t edges

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Preferential attachment model

Barabasi, 1999

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Preferential attachement

ki (t) = m0

( ti

)1/2

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Preferential attachement

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Preferential attachment

Time evolution of a node degree

ki (t) = m0

( ti

)1/2

Degree distribution function:

P(k) =2m2

0

k3

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Preferential attachement

Preferential attachment vs random graph

BA : P(k) =2m2

0

k3, ER : P(k) =

〈k〉ke−〈k〉

k!, 〈k〉 = pn

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Preferential attachement

Preferential attachment vs random graph

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Preferential attachment model

m0 = 1 m0 = 2 m0 = 3

igraph: barabasi.game()

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Preferential attachment model

Node degree distribution - power law):

P(k) =2m2

0

k3

Average path length :

〈L〉 ∼ log(N)/ log(log(N))

Clustering coefficient (numerical result):

C ∼ N−0.75

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Small world

Motivation: keep high clustering, get small diameter

Clustering coefficient C = 1/2Graph diameter d = 8

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Small world

Watts and Strogatz, 1998

Single parameter model, interpolation between regular lattice and randomgraph

start with regular lattice with n nodes, k edges per vertex (nodedegree), k << n

randomly connect with other nodes with probability p, forms pnk/2”long distance” connections from total of nk/2 edges

p = 0 regular lattice, p = 1 random graph

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Small world

Watts, 1998

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Small world model

20% rewiring:ave. path length = 3.58 → ave. path length = 2.32clust. coeff = 0.49 → clust. coeff = 0.19

igraph:watts.strogatz.game()

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Small world model

Node degree distribution function - Poisson like (numerical result)

Average path length (analytical result) :

〈L〉 ∼ log(N)

Clustering coefficientC = const

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Model comparison

Random BA model WS model Empirical networks

P(k) λke−λ

k! k−3 poisson like power lawC 〈k〉/N N−0.75 const large

〈L〉 log(N)log(〈k〉)

log(N)log log(N) log(N) small

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References

On random graphs I, P. Erdos and A. Renyi, PublicationesMathematicae 6, 290297 (1959).

On the evolution of random graphs, P. Erdos and A. Renyi,Publicaton of the Mathematical Institute of the Hungarian Academyof Sciences, 17-61 (1960)

Collective dynamics of small-world networks. Duncan J. Watts andSteven H. Strogatz. Nature 393 (6684): 440-442, 1998

Emergence of Scaling in Random Networks, A.L. Barabasi and R.Albert, Science 286, 509-512, 1999

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