Grand Canonical Approach to an Interacting Network

Abstract

We consider a network composed of an arbitrary number of directed links. We employ a grand canonical partition function to study the statistical averages of the network in equilibrium. The Hamiltonian is composed of two parts: a “free” Hamiltonian H0 attributing a constant energy E to each link, and an interacting Hamiltonian Hint involving terms quadratic in the number of links. A Gaussian integration leads to a reformulated Hamiltonian, where now the number of links appears linearly. The reformulated Hamiltonian allows obtaining the exact behavior in limiting cases. At high temperatures the system reproduces the behavior of the free model, while at low temperatures the thermodynamic behavior is obtained by using a renormalized chemical potential, μeff = μ + l, where l is the strength of the interaction. We also resort to a mean field approximation, describing accurately the system over the entire range of all dynamical parameters.