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000009986 084__ $$2WoS$$aPhysics, Fluids & Plasmas
000009986 084__ $$2WoS$$aPhysics, Mathematical
000009986 1001_ $$0P:(DE-HGF)0$$aFoster, D.$$b0
000009986 245__ $$aCommunities, clustering phase transitions, and hysteresis: Pitfalls in constructing network ensembles
000009986 260__ $$aCollege Park, Md.$$bAPS$$c2010
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000009986 520__ $$aEnsembles 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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000009986 7001_ $$0P:(DE-HGF)0$$aFoster, J.$$b1
000009986 7001_ $$0P:(DE-HGF)0$$aPaczuski, M.$$b2
000009986 7001_ $$0P:(DE-Juel1)136887$$aGrassberger, P.$$b3$$uFZJ
000009986 77318 $$2Crossref$$3journal-article$$a10.1103/physreve.81.046115$$bAmerican Physical Society (APS)$$d2010-04-27$$n4$$p046115$$tPhysical Review E$$v81$$x1539-3755$$y2010
000009986 773__ $$0PERI:(DE-600)2844562-4$$a10.1103/PhysRevE.81.046115$$gVol. 81, p. 046115$$n4$$p046115$$q81<046115$$tPhysical review / E$$v81$$x1539-3755$$y2010
000009986 8567_ $$uhttp://dx.doi.org/10.1103/PhysRevE.81.046115
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