Communities, clustering phase transitions, and hysteresis: Pitfalls in constructing network ensembles

Foster, D.; Grassberger, P.; Paczuski, M.; Foster, J.

doi:10.1103/PhysRevE.81.046115

Journal Article

PreJuSER-9986

Communities, clustering phase transitions, and hysteresis: Pitfalls in constructing network ensembles

Foster, D. ; Foster, J. ; Paczuski, M. ; Grassberger, P.FZJ*

2010
APS College Park, Md.

Physical review / E 81(4), 046115 (2010) [10.1103/PhysRevE.81.046115]

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Please use a persistent id in citations: http://hdl.handle.net/2128/9313 doi:10.1103/PhysRevE.81.046115

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 "biased rewiring model" where clustering is enhanced by means of a fugacity as in the Strauss (or "triangle") model, but where the number of links attached to each node is strictly preserved. Similar models have been proposed previously in Milo [Science 298, 824 (2002)]. 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 nonregular networks that we studied (including Erdoumls-Reacutenyi, scale-free, and several real-world networks) multiple jumps resembling first order transitions appear. The jumps coincide with the sudden emergence of "cluster cores:" groups of highly interconnected nodes with higher than average degrees, where each edge participates in many triangles. Hence, clustering is not smoothly distributed throughout the network. Once formed, the cluster cores are difficult to remove, leading to strong hysteresis. To study the cluster cores visually, we introduce q -clique adjacency plots. Cluster cores constitute robust communities that emerge spontaneously from the triangle generating process, rather than being put explicitly into the definition of the model. All the quantities we measured including the modularity, assortativity, clustering and number of four and five-cliques exhibit simultaneous jumps and are equivalent order parameters. 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 strong hysteresis which leads to broken ergodicity on realistic sampling time scales.

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