An alternative approach to determining average distance in a class of scale-free modular networks

TL;DR

This paper analytically derives a closed-form expression for the average distance in a class of scale-free modular networks, revealing their small-world nature and the impact of their architecture.

Contribution

It introduces a new analytical method to compute average distance in deterministic scale-free modular networks, confirming small-world properties.

Findings

Average distance scales logarithmically with network size

Networks exhibit small-world behavior due to their architecture

Analytical results match numerical simulations

Abstract

Various real-life networks of current interest are simultaneously scale-free and modular. Here we study analytically the average distance in a class of deterministically growing scale-free modular networks. By virtue of the recursive relations derived from the self-similar structure of the networks, we compute rigorously this important quantity, obtaining an explicit closed-form solution, which recovers the previous result and is corroborated by extensive numerical calculations. The obtained exact expression shows that the average distance scales logarithmically with the number of nodes in the networks, indicating an existence of small-world behavior. We present that this small-world phenomenon comes from the peculiar architecture of the network family.