Author(s)

A. Nicolaidis^1*, K. Kosmidis^2,3, V. Kiosses¹

¹Theoretical Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece.
²Computing & Communications Center, Aristotle University of Thessaloniki, Thessaloniki, Greece.
³School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, Bremen, Germany.

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 H_int 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, μ_eff = 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. A detailed Monte-Carlo simulation verifies our theoretical expectations. We indicate that our model may serve as a prototype model to address a number of different systems.

Complex Networks, Statistical Mechanics, Monte Carlo Simulations, Grand Canonical Ensemble

Nicolaidis, A. , Kosmidis, K. and Kiosses, V. (2015) Grand Canonical Approach to an Interacting Network. Journal of Modern Physics, 6, 472-482. doi: 10.4236/jmp.2015.64051.