Random Graph models

Presenter: Ildar NurgalievLab: Dainfos

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Random Graphmodels

1. Erd�os�R�enyi model

2. Watts and Strogatz model

3. Barab�asi�Albert

4. Chung-Lu model

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Erd�os�R�enyi model (ER)concepts

2 variants:

1. G(n, M)

2. G(n, p)

1) a graph is chosen uniformly at random from thecollection of all graphs which have n nodes and M edges.2) a graph is constructed by connecting nodes randomly:

Expected N of edges(n

2

)p ≈ n2p →∞ (G (n, p) ≈ G (n,M))

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Erd�os�R�enyi model (ER)properties of G(n,p)

• on average(N

2

)edges

• thus distribution of degree is binomial:P(deg(v) = k) =

(n−1

k

)pk(1− p)n−1−k

• graph does not have heavy tails, thus has low clustering

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Erd�os�R�enyi model (ER)properties of G(n,p)

- if np < 1 then no connected component of size > p(log(n))- if np = 1 then almost have component whose size ≈ n2/3

- if np → c > 1 then unique giant component

ln(n)

n− sharp threshold

- if p < (1− ε) ln(n)n

then contains isolated vertices

- if p > (1− ε) ln(n)n

then a graph is surely connected

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Watts and Strogatz modelconcepts

produces graphs with properties:

• small-world properties

• including short average path lengths

• high clustering

clustering coe�cient could lead to triadic closure property:A-B and A-C → B-C (strong or weak tie)

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Watts and Strogatz modelInput

Three parameters:

• N

• K - links/node

• β - prob of connection random pair of nodes, for eachlink in graph

Out: undirected graph with N nodes andNK

2edges

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Watts and Strogatz modelAlgorithm

• construct a regular ring lattice:

• each of N connected to K neighbors: K/2 on each side

• if 0 < |i − j |mod(N − 1− K

2) ≤ K

2then edge occur

• rewire:

• for every node take every (ni , nj) with i < j andrewire it with β

• replace (ni , nj) with (ni , nk) (k - uniform probability)• k 6= i and no dublication

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Watts and Strogatz modelAlgorithm

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Watts and Strogatz modelProperties

• interpolite β

• β = 0: regular lattice• β = 1: approaching G(n,p) (ER-graph)

n = N and p =NK

2(N2

)• avg path length: `(0) = N/2K >> 1 and `(1) =

ln(N)

ln(K )

• clustering coef: C (0) =3(K − 2)

4(K − 1)and C (1) =

K

N

• degree distr

• β → 0 - Dirac delta function• β → 1 - Poisson distribution

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Watts and Strogatz modelLimitations

• This model implies a �xed number of nodes and thuscannot be used to model network growth

• It produces an unrealistic degree distribution

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Barab�asi�Albert modelConcepts

Scale-free networks using a preferential attachmentmechanism but low clustering coe�cient

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Barab�asi�Albert modelAlgorithm

• The network begins with an initial connected networkof m0 nodes.

• new node connect to m ≤ m0 existing

• pi =ki∑j kj

(new node connected to i)

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Barab�asi�Albert modelProperties

• avg path length: ` ∼ ln(N)

lnln(N)

• empirical results: C ∼ N−0.75 (dependent on systemsize)

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Chung-Lu modelAlgorithm

• Given expected degree sequence w = (w1,w2, ...,wn)

• pij of an edge is proportional to the wiwj (with loops)

• pij =wiwj∑k wk

and maxi w2

i <∑k

wk

• ps: G(n,p) has degree sequence (pn,pn,...,pn)

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The End

Thank you!

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