An ǫ-expansion for Small-World Networks

– I construct a well-defined expansion in ǫ = 2− d for diffusion processes on smallworld networks. The technique permits one to calculate the average over disorder of moments of the Green’s function, and is used to calculate the average Green’s function and fluctuations to first non-leading order in ǫ, giving results which agree with numerics. This technique is also applicable to other problems of diffusion in random media. The small-world network [1] has served as a fundamental model in the field of networks [2]. However, the problem of averaging over the possible different random connections in the smallworld network is severe in low dimensions: a study [3] of the properties of even the simple problem of diffusion on the one-dimensional network leads to a difficult problem that, thus far, has only been tackled approximately. The physical reason for this problem is a breakdown of mean-field theory [4] in dimensions d less than two, and the emergence of strong site-to-site fluctuations of the Green’s function, so that the properties of the system cannot be represented by simply studying the average. However, this opens the possibility of perturbing in ǫ = 2− d, as will be shown in this paper. The small-world network is constructed by starting with a regular lattice in d-dimensions. Then, some set of long-range links are added: a given pair of sites i, j is connected with probability pa/V , where V is the total number of sites in the system. Here, we define a length a as the lattice scale, and p as the density of links. Then, as V → ∞, each site has a Poisson distribution of links emanating from it, with on average pa links. Typically, the links, if any, leaving a given site will connect that site to other sites far away in the system. Looking for universal results, we consider the case of a low density of links, pa << 1. Ignoring sample-to-sample fluctuations, the natural mean-field system to consider is one in which each site is coupled to all others with a strength ∼ p/V . This leads to a solvable problem with a correlation length ξ ∝ p, or a correlation volume ξ ∝ p. Returning to the original problem with fixed links, we see that such a volume has p links in it, and as p → 0, this number of links tends to zero for p ≤ 2. This is a major problem. Mean-field theory ignores fluctuations in the number of links, which is only justified if the number of links is large, a condition which is not satisfied in this case for d < 2. This problem is a result of a violation of the modified Harris criterion introduced in [5]. Instead, we expect that the correlation length for the average Green’s function must be at least p for small p, so that there is on average at least one link in a correlation volume. We