It’s a Small World by Jamie Luo

Introduction Small World Networks and their place in Network Theory An application of a 1D small world network to model the spread of an infection (Cristopher Moore and M.E.J. Newman)

Random Graphs In 1959 Erdos and Renyi define a random graph as N labelled nodes connected by n edges, which are chosen randomly from the N(N21)/2 possible edges. Eg:Below are cases for N=10 with n=0 and n=7

Regular Lattices On the other extreme you have regular lattices. Eg:Z 2 or the one dimensional lattice depicted below

Some Definitions For a G graph of n vertices labelled v 1,..., v n : Characteristic path length, l(G). l is defined as the number of edges in the shortest path between two vertices, averaged over all pairs of vertices. Clustering coefficient, C(G).

A Small World Network In 1998 Duncan J. Watts & Steven H. Strogatz produce a new network model with a parameter 0 < p < 1, that is regular lattice at φ=0 but is a random graph at p=1.

Crossover l ~ L, linearly on a regular lattice, where L=linear dimension l α log(N)/log(z), for a random graph, where N=the number of sites and z=the average degree of the vertices The small world model lies yet again between these two extremes. If we fix p then: For small N, l(N,p) ~ L linearly For large enough N, l(N,p) ~ log(N) It turns out there is a crossover from the small world to a ‘large one’.

Scaling Similar to the correlation length behaviour in statistical mechanics, at some intermediate system value N = l, where the transition occurs, we expect, l ∼ p -τ. Additionally, close to the transition point, l(N, p) should obey the finite- size scaling relation: where f(u) is a universal scaling function obeying, f (u) ∼ u if u << 1 f (u) ∼ ln u if u>>1 It has been analytically demonstrated that for this model τ=1.

Another Small World To investigate the spread of an epidemic infection we make an alteration to the construction of Strogatz and Watts’ model. Instead of rewiring edges we simply add shortcuts between vertices with a probability φ for each bond.

An Infectious Model The two parameters we are interested in for this model of the spread of an infectious disease are susceptibility, the probability that an individual in contact with a disease will contract it and transmissibility, the probability that contact between an infected individual and a healthy but susceptible individual will result in the latter contracting the disease. To deal with susceptibility and transmissibility you can incorporate into the model site and bond percolation respectively. We will deal with the site percolation case in detail. So take our small world and say any individual is assumed to be susceptible with probability p. Then we just fill the sites in our small world (the individuals ) to indicate they are susceptible with probability p.

Idea We always start with one infected individual from which the disease spreads. We partition our model into ‘local clusters’ which are those collections of connected filled sites before the shortcuts are introduced. All sites in the local cluster containing the infected individual are infected immediately. Then in the next time step every local cluster connected to the infected cluster by a single step along a shortcut is then infected and this infected cluster grows accordingly.

Analysis We know that the probability that two random sites are connected by a shortcut is, The approximation is true for sufficiently large L. For k=1, the average number of local clusters of length i is,

Define v to be a vector at each time step, with v i = the probability that a local cluster of size i has just been added to the infected cluster. This is our means for constructing the infected cluster. We want to know v’ from v. At or below the percolation threshold, the v i are small and so the v i ‘ depend linearly on the v i according to a transition matrix M and thus, where,

Eigenvalues Consider the largest eigenvalue of M, call it λ. Case 1: λ<1, then v tends to 0. Case 2: λ>1then v grows to towards the size of the system. Thus the percolation threshold occurs at λ=1. Generally finding λ is difficult but for large L we can approximate M by, which is the outer product of two vectors. So we can rewrite, with the eigenvectors of M have the form,

pcpc This simplifies to give, and thus for k=1 Setting λ=1 allows us to derive values for,

More Results For general k, and at the threshold then, which implies that p c is then the root of a k+1 order polynomial. For bond percolation we have an analogous scenario to site percolation for the case k=1. For general k there is a method to find solutions but this becomes very tedious for larger k. Numerical results exist in the case when both site and bond percolation are allowed but no analytical solutions.

Summary Small Worlds interpolate between the extreme models of random graphs and regular lattices They exhibit the shorter characteristic length scales of random graphs and the higher clustering coefficients regular lattices Thus they can be applied usefully to model real world networks which share these properties.