Phases of Small Worlds: A Mean Field Formulation

TL;DR

This paper introduces a mean field approach to analytically derive the small-world property in various random network models, revealing phase structures and conditions for small-world behavior.

Contribution

It provides a novel mean field formulation for small-world networks, extending to dimorphic networks and linking network properties to phase transitions.

Findings

Logarithmic scaling of shortest paths in Erdős-Rényi networks

Mean field approximation maps to a spin chain problem

Small world regimes can be induced in sub-networks

Abstract

A network is said to have the properties of a small world if a suitably defined average distance between any two nodes is proportional to the logarithm of the number of nodes, $N$. In this paper, we present a novel derivation of the small-world property for Gilbert-Erd\"os-Renyi random networks. We employ a mean field approximation that permits the analytic derivation of the distribution of shortest paths that exhibits logarithmic scaling away from the phase transition, inferable via a suitably interpreted order parameter. We begin by framing the problem in generality with a formal generating functional for undirected weighted random graphs with arbitrary disorder, recovering the result that the free energy associated with an ensemble of Gilbert graphs corresponds to a system of non-interacting fermions identified with the edge states. We then present a mean field solution for this model and extend it to more general realizations of network randomness. For a two family class of stochastic block models that we refer to as dimorphic networks, which allow for links within the different families to be drawn from two independent discrete probability distributions, we find the mean field approximation maps onto a spin chain combinatorial problem and again yields useful approximate analytic expressions for mean path lengths. Dimorophic networks exhibit a richer phase structure, where distinct small world regimes separate in analogy to the spinodal decomposition of a fluid. We find that is it possible to induce small world behavior in sub-networks that by themselves would not be in the small-world regime.