Geometric evolution of complex networks

TL;DR

This paper introduces a geometric network growth model using homogeneous attachment and Fermi-Dirac connection probability, allowing control over degree distribution, correlations, and clustering, revealing a phase transition based on temperature parameter.

Contribution

It provides an analytical framework for geometric network evolution with tunable degree distributions and correlations, incorporating geometry and phase transition phenomena.

Findings

Degree distribution can be tailored to any form, including scale-free.

Degree correlations can be tuned to be assortative or disassortative.

A phase transition in clustering coefficient occurs at a critical temperature.

Abstract

We present a general class of geometric network growth mechanisms by homogeneous attachment in which the links created at a given time $t$ are distributed homogeneously between a new node and the exising nodes selected uniformly. This is achieved by creating links between nodes uniformly distributed in a homogeneous metric space according to a Fermi-Dirac connection probability with inverse temperature $\beta$ and general time-dependent chemical potential $\mu(t)$. The chemical potential limits the spatial extent of newly created links. Using a hidden variable framework, we obtain an analytical expression for the degree sequence and show that $\mu(t)$ can be fixed to yield any given degree distributions, including a scale-free degree distribution. Additionally, we find that depending on the order in which nodes appear in the network---its $\textit{history}$---the degree-degree correlation can be tuned to be assortative or disassortative. The effect of the geometry on the structure is investigated through the average clustering coefficient $\langle c \rangle$. In the thermodynamic limit, we identify a phase transition between a random regime where $\langle c \rangle \rightarrow 0$ when $\beta < \beta_\mathrm{c}$ and a geometric regime where $\langle c \rangle > 0$ when $\beta > \beta_\mathrm{c}$.