2. Random Graphs

The Erdös-Rényi models

Distinguish:– Equilibrium random networks– Nonequilibrium random networks

Equilibrium random networks

A classical undirected random graph:– the total number of vertices is fixed– connect randomly chosen pairs of vertices

Nonequilibrium random network

A classical random graph that grows through simultaneous addition of vertices and links– at each time step a new vertex is added– simultaneously, a pair of randomly chosen vertices is

connected

Status

Graph theory:– Equilibrium networks with a Poisson degree

distribution

Physics:– Nonequilibrium (growing networks), percolation

Statistical sense

A particular observed network is only one member of a statistical ensemble of all possible realizationsRandom network -> Statistical ensembleN nodes -> How should we understand the degree distribution? It determines the the ensemble of the equilibrium random networks

2/)1(2 NNpossible graphs

The Erdos-Renyi model

Definition: N labeled nodes connected by n links which are chosen randomly from the N(N-1)/2 possible linksThere are graphs with N nodes and n links

n

NN 2/)1(

Alternative definition

Binomial model: start with N nodes, every pair of nodes being connected with probability pThe total number of links, n, is a random variable– E(n)=pN(N-1)/2

Probability of generating a graph, G0{N,n}

nNN

n ppGP

2

)1(

0 )1()(

Growing a graph

Sometimes we will study properties of the graph as p increasesAssign a random number qi[0,1] to attach links and then links appear as p is increased p> q i

We are interested in the “static” properties of the graph when N-> and keeping constant p or n

N->

Definition: almost every graph has a property Q if the probability of having Q approaches 1 as N-> The main goal of Random Graph theory is to determine at what connection probability p a particular property of a graph most likely arises

Many important properties appear suddenly:– almost everygraph has the property– almost no graph has it

Usually there exists a critical probability pc(N)

p(N) probability that almost every graph has property Q

If p(N) grows slower than pc(N)

If p(N) grows faster than pc(N)

]0)(/)([ Nc NpNp

0)(,lim

QP pNN

])(/)([ Nc NpNp

1)(,lim

QP pNN

Examples

Larger graphs with the same p contain more links since n=pN(N-1)/2– Appearance of cycles can occur for smaller p in large

graphs than in smaller ones [pc(N->)->0]

Average degree of the graph

pNNpNnk )1(/2

Subgraphs

P1 set of nodes, E1 set of links

G1(P1,E1) is a subgraph of G(P,E) if all nodes of G1 belong to G and links too.

Basic subgraphs:– cycles– trees– complete graphs

Evolution of the graph (p grows)

pNGG ,

Subgraph in graph

F small graph of k nodes and l linksHow many subgraphs like F exist in G?

This expected value depends on p. If N>>k

lpa

k

k

NXE

!)(

a

pNXE

lk

)(

There are no subgraphs like FIf = constant => mean number of subgraphs is a finite numberThe critical probability

0)(0)( if / XENNpN

lk

lkc cNNp /)(

Tree of order k: l=k-1Cycle of order k: l=kComplete subgraph l=k(k-1)/2

We can see how the subgraphs appear when increasing p

Mean connectivity

<k>=pNIf then <k> is a constantIf 0< <k> < 1 almost surely all clusters are either trees or clusters containing exactly one cycleAt <k>=1 the structure changes abruptly. Cycles appear and a giant cluster develops

1 Np

Degree distribution

The degree of a node follows a binomial distribution (in a random graph with p)

Probability that a given node has a connectivity kFor large N, Poisson distribution

kNki pp

k

NkkP

1)1(

1)(

!!

)()(

k

ke

k

pNekP

kk

kpN

Mean short path

Assume that the graph is homogeneousThe number of nodes at distance l are <k> l

How to reach the rest of the nodes?lrand to reach all nodes => kl=N

pN

N

k

Nl

ln

ln

ln

lnrand

Clustering coefficient

Probability that two nodes are connected (given that they are connected to a third)?

N

kpCrand

Nk

Crand 1

while it is constant for real networks

Spectrum: random matrices

If Aij real, symmetric, NxN uncorrelated random matrix <Aij>=0 and <Aij

2>=2

Density of eigenvalues of

Wigner’s or semicircle law (late 50’s)

seN

otherwi0

2if)2(

4)( 2

22

NA /

Spectrum: random graph

<Aij>= not 0

2 =p(1-p)Plotting

() semicircel law as N increases (p constant)

)1( vs)1(

pNppNp

In general,z<1:– semicircle law– exists an infinite cluster 1 (principal, largest) is isolated, grows like N

z<1: most of the graphs are trees (odd moment vanish). The spectral density contains the weighted sum of the spectral densities of all finite graphs

zcNNp )(

Generalized random graphs

One can construct a graph introducing the degree distribution as an input

How do the properties of the network change with the exponent? decreases from to 0

kkP )(

<k>=kmax-+2 (kmax <N, max degree)

The infinite cluster emerges when

There exists a value 0=3.47875.....

>0 disconnected

>0 almost surely connected

1

0)()2(k

KPkkQ

Exponential cutoff (observed in real world networks)Normalitzable for any k>2 disconnected<2 connected

1 )( / keCkkP k

NON-RANDOM aspects of the topology of real networks

Growing networks

See hand-written notes

Scaling