Number of loops of size h in growing scale-free networks

TL;DR

This paper analytically and numerically investigates how the number of loops of size h scales with system size in various growing scale-free networks, revealing topological phase transitions.

Contribution

It provides the first analytic expression for the scaling of loop counts in the BA network and compares it with other models, highlighting topological phase transitions.

Findings

Analytic expression for $N_h(t)$ in BA networks

Scaling laws change at topological phase transitions

Numerical simulations confirm the theoretical predictions

Abstract

The hierarchical structure of scale-free networks has been investigated focusing on the scaling of the number $N_h(t)$ of loops of size h as a function of the system size. In particular we have found the analytic expression for the scaling of $N_h(t)$ in the Barab\'asi-Albert (BA) scale-free network. We have performed numerical simulations on the scaling law for $N_h(t)$ in the BA network and in other growing scale free networks, such as the bosonic network (BN) and the aging nodes (AN) network. We show that in the bosonic network and in the aging node network the phase transitions in the topology of the network are accompained by a change in the scaling of the number of loops with the system size.