Complex interactions call for the sharing of information between different entities. In a recent paper, we introduced a combinatoric model which concretizes this idea via a string-matching rule. The model was shown to lend itself to analysis regarding certain topological features of the network. In this paper, we will introduce a statistical physics description of this network in terms of a Potts model. We will give an explicit mean-field treatment of a special case that has been proposed as a model for gene regulatory networks, and derive closed-form expressions for the topological coefficients. Simulations of the hidden variable network are then compared with numerically integrated results.(c) 2007 American Institute of Physics.