  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.
