Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles

TL;DR

This paper investigates phase transitions and hysteresis in network models that preserve degree sequences while enhancing clustering, revealing pitfalls in using such models as null models for real-world networks due to strong hysteresis effects.

Contribution

The study demonstrates phase transitions and hysteresis in clustering-enhanced network models with fixed degree sequences, highlighting issues in their application as null models.

Findings

Regular graphs with degree k > 2 show a single first order transition.

Non-regular networks exhibit multiple jumps and hysteresis.

Cluster cores emerge as communities, causing modeling pitfalls.

Abstract

Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a model where clustering is enhanced by means of a fugacity term as in the Strauss (or "triangle") model, but where the degree sequence is strictly preserved -- thus maintaining the quenched heterogeneity of nodes found in the original degree sequence. Similar models had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We find that our model exhibits phase transitions as the fugacity is changed. For regular graphs (identical degrees for all nodes) with degree k > 2 we find a single first order transition. For all non-regular networks that we studied (including Erdos - Renyi and scale-free networks) we find multiple jumps resembling first order transitions, together with strong hysteresis. The latter transitions are driven by the sudden emergence of "cluster cores": groups of highly interconnected nodes with higher than average degrees. To study these cluster cores visually, we introduce q-clique adjacency plots. We find that these cluster cores constitute distinct communities which emerge spontaneously from the triangle generating process. Finally, we point out that cluster cores produce pitfalls when using the present (and similar) models as null models for strongly clustered networks, due to the very strong hysteresis which effectively leads to broken ergodicity on realistic time scales.