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@ARTICLE{Foster:9986,
      author       = {Foster, D. and Foster, J. and Paczuski, M. and Grassberger,
                      P.},
      title        = {{C}ommunities, clustering phase transitions, and
                      hysteresis: {P}itfalls in constructing network ensembles},
      journal      = {Physical review / E},
      volume       = {81},
      number       = {4},
      issn         = {1539-3755},
      address      = {College Park, Md.},
      publisher    = {APS},
      reportid     = {PreJuSER-9986},
      pages        = {046115},
      year         = {2010},
      note         = {Record converted from VDB: 12.11.2012},
      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.},
      keywords     = {J (WoSType)},
      cin          = {JSC},
      ddc          = {530},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {Scientific Computing (FUEK411) / 411 - Computational
                      Science and Mathematical Methods (POF2-411)},
      pid          = {G:(DE-Juel1)FUEK411 / G:(DE-HGF)POF2-411},
      shelfmark    = {Physics, Fluids $\&$ Plasmas / Physics, Mathematical},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000277265900018},
      doi          = {10.1103/PhysRevE.81.046115},
      url          = {https://juser.fz-juelich.de/record/9986},
}