Statistical mechanical approach of complex networks with weighted links

TL;DR

This paper models complex networks with weighted links using a statistical mechanical approach, analyzing how weights and spatial factors influence network growth and degree distributions.

Contribution

It introduces a weighted network growth model incorporating spatially-dependent preferential attachment and analyzes the resulting degree and energy distributions.

Findings

Connectivity distribution converges to weight distribution under certain conditions.

Network properties depend on the spatial interaction parameter $oldsymbol{\alpha_A/d}$.

Model captures transition from short-range to long-range interactions.

Abstract

Systems which consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social systems. We numerically analyze the growth of $d$-dimensional geographic networks (characterized by the index $\alpha_G\geq0$; $d = 1, 2, 3, 4$) whose links are weighted through a predefined random probability distribution, namely $P(w) \propto e^{-|w - w_c|/\tau}$, $w$ being the weight $ (w_c \geq 0; \; \tau > 0)$. In this model, each site has an evolving degree $k_i$ and a local energy $\varepsilon_i \equiv \sum_{j=1}^{k_i} w_{ij}/2$ ($i = 1, 2, ..., N$) that depend on the weights of the links connected to it. Each newly arriving site links to one of the pre-existing ones through preferential attachment given by the probability $\Pi_{ij}\propto \varepsilon_{i}/d^{\,\alpha_A}_{ij} \;\;(\alpha_A \ge 0)$, where $d_{ij}$ is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to $\alpha_A/d>1$ and $0\leq \alpha_A/d \leq 1$; $\alpha_A/d \to \infty$ corresponds to interactions between close neighbors, and $\alpha_A/d \to 0$ corresponds to infinitely-ranged interactions. The site energy distribution $p(\varepsilon)$ corresponds to the usual degree distribution $p(k)$ as the particular instance $(w_c,\tau)=(2,0)$. We numerically verify that the corresponding connectivity distribution $p(\varepsilon)$ converges, when $\alpha_A/d\to\infty$, to the weight distribution $P(w)$ for infinitely narrow distributions (i.e., $\tau \to \infty, \,\forall w_c$) as well as for $w_c\to0, \, \forall\tau$.