Relative Canonical Network Ensembles -- (Mis)characterizing Small-World Networks

TL;DR

This paper introduces Relative Canonical Network ensembles to analyze the generic features of networks with specific properties, revealing that high small-world-ness does not necessarily characterize small-world networks, and identifying key features like hubs.

Contribution

The paper proposes a novel framework for characterizing network properties by comparing them to the most generic networks with those properties, challenging traditional notions of small-world networks.

Findings

High small-world-ness does not characterize small-world networks.

Hubs and cliques emerge as features in less generic networks.

Average shortest path length and Euclidean link length better characterize small-world networks.

Abstract

What do generic networks that have certain properties look like? We define Relative Canonical Network ensembles as the ensembles that realize a property R while being as indistinguishable as possible from a generic network ensemble. This allows us to study the most generic features of the networks giving rise to the property under investigation. To test the approach we apply it first to the network measure "small-world-ness", thought to characterize small-world networks. We find several phase transitions as we go to less and less generic networks in which cliques and hubs emerge. Such features are not shared by typical small-world networks, showing that high "small-world-ness" does not characterize small-world networks as they are commonly understood. On the other hand we see that for embedded networks, the average shortest path length and total Euclidean link length are better at characterizing small-world networks, with hubs that emerge as a defining feature at low genericity. We expect the overall approach to have wide applicability for understanding network properties of real world interest.