TITLE:
Grand Canonical Approach to an Interacting Network
AUTHORS:
A. Nicolaidis, K. Kosmidis, V. Kiosses
KEYWORDS:
Complex Networks, Statistical Mechanics, Monte Carlo Simulations, Grand Canonical Ensemble
JOURNAL NAME:
Journal of Modern Physics,
Vol.6 No.4,
March
27,
2015
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. 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.