It’s a Small World

1. It’s a Small World by Jamie Luo

2. 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)

3. Random Graphs • In 1959 Erdosand 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

4. Regular Lattices • On the other extreme you have regular lattices. • Eg: Z2 or the one dimensional lattice depicted below

5. Some Definitions • For a G graph of n vertices labelled v1, ..., vn: • 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).

6. 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.

9. 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 fixpthen: • For small N, l(N,p) ~ L linearly • For large enough N, l(N,p) ~ log(N) • It turns out there is a crossoverfrom the small world to a ‘large one’.

10. 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 ifu << 1 • f (u) ∼ lnu if u>>1 • It has been analytically demonstrated that for this model τ=1.

11. 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.

12. 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 susceptibilityand transmissibilityyou 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.

14. 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.

15. 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 iis,

16. Define v to be a vector at each time step, with vi = 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 vi are small and so the vi ‘ depend linearly on the vi according to a transition matrix M and thus, where,

17. Eigenvalues • Consider the largest eigenvalue of M, call it λ. • Case 1: λ<1, then v tends to 0. • Case 2: λ>1 then v grows to towards the size of the system. • Thus the percolation threshold occurs at λ=1. Generally finding λis 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,

18. pc • This simplifies to give, and thus for k=1 • Setting λ=1 allows us to derive values for,

19. More Results • For general k, and at the threshold then, which implies that pc 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 siteandbondpercolation are allowed but no analytical solutions.

20. 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.