The Erdös-Rényi models 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: Physics: 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 realizations Random network -> Statistical ensemble N nodes -> How should we understand the degree distribution? It determines the the ensemble of the equilibrium random networks possible graphs

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

Alternative definition Binomial model: start with N nodes, every pair of nodes being connected with probability p The total number of links, n, is a random variable E(n)=pN(N-1)/2 Probability of generating a graph, G0{N,n}

Growing a graph Sometimes we will study properties of the graph as p increases Assign a random number qi[0,1] to attach links and then links appear as p is increased p> qi 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)

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 No clar. Segons BA té un valor crític que és independent del tamany del sistema

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)

Subgraph in graph F small graph of k nodes and l links How many subgraphs like F exist in G? This expected value depends on p. If N>>k

There are no subgraphs like F If = constant => mean number of subgraphs is a finite number The critical probability

Tree of order k: l=k-1 Cycle of order k: l=k Complete subgraph l=k(k-1)/2 We can see how the subgraphs appear when increasing p

Mean connectivity <k>=pN If then <k> is a constant If 0< <k> < 1 almost surely all clusters are either trees or clusters containing exactly one cycle At <k>=1 the structure changes abruptly. Cycles appear and a giant cluster develops

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 k For large N, Poisson distribution

Mean short path Assume that the graph is homogeneous The number of nodes at distance l are <k>l How to reach the rest of the nodes? lrand to reach all nodes => kl=N

Clustering coefficient Probability that two nodes are connected (given that they are connected to a third)? while it is constant for real networks

Spectrum: random matrices If Aij real, symmetric, NxN uncorrelated random matrix <Aij>=0 and <Aij2>=2 Density of eigenvalues of Wigner’s or semicircle law (late 50’s)

Spectrum: random graph <Aij>=  not 0 2 =p(1-p) Plotting () semicircel law as N increases (p constant)

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

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

<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

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

NON-RANDOM aspects of the topology of real networks

Growing networks See hand-written notes

Scaling