Phases of Small Worlds: A Mean Field Formulation

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