Complex network growth model: Possible isomorphism between nonextensive statistical mechanics and random geometry

TL;DR

This paper presents numerical evidence of an isomorphism between nonextensive statistical mechanics, specifically the energy q-exponential distribution, and a random geometric growth model based on preferential attachment with weighted links.

Contribution

It demonstrates a novel connection between nonextensive entropy optimization and a specific random network growth model, expanding the understanding of their relationship.

Findings

Numerical evidence of isomorphism between nonextensive entropy and random network growth.

Identification of specific parameters ($q=4/3$, $eta_q o 10/3$) for the distribution.

Connection to preferential attachment models with exponential weights.

Abstract

In the realm of Boltzmann-Gibbs statistical mechanics there are three well known isomorphic connections with random geometry, namely (i) the Kasteleyn-Fortuin theorem which connects the $\lambda \to 1$ limit of the $\lambda$-state Potts ferromagnet with bond percolation, (ii) the isomorphism which connects the $\lambda \to 0$ limit of the $\lambda$-state Potts ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism which connects the $n \to 0$ limit of the $n$-vector ferromagnet with self-avoiding random walk in linear polymers. We provide here strong numerical evidence that a similar isomorphism appears to emerge connecting the energy $q$-exponential distribution $\propto e_q^{-\beta_q \varepsilon}$ (with $q=4/3$ and $\beta_q \omega_0 =10/3$) optimizing, under simple constraints, the nonadditive entropy $S_q$ with a specific geographic growth random model based on preferential attachment through exponentially-distributed weighted links, $\omega_0$ being the characteristic weight.