Phase Transitions in an Aging Network

TL;DR

This paper investigates how different attachment rules in an aging network influence its structural properties, revealing a phase diagram where scale-free behavior occurs only along a specific line, while small-world features are widespread.

Contribution

It introduces a generalized attachment probability incorporating degree and age, and maps the resulting phase diagram showing conditions for scale-free and small-world properties.

Findings

Scale-free degree distribution occurs only along a specific line in the phase diagram.

Small-world property exists over a broad region in the phase diagram.

The network's behavior depends critically on the parameters controlling attachment probability.

Abstract

We consider a growing network in which an incoming node gets attached to the $i^{th}$ existing node with the probability $\Pi_i \propto {k_i}^{\beta}\tau_i^{\alpha}$, where $k_{i}$ is the degree of the $i^{th}$ node and $\tau_i$ its present age. The phase diagram in the ${{\alpha}-{\beta}}$ plane is obtained. The network shows scale-free behaviour, i.e., the degree distribution $P(k) \sim k^{-\gamma}$ with $\gamma =3$ only along a line in this plane. Small world property, on the other hand, exists over a large region in the phase diagram.