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  <doc>
    <id>5573</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2015-08-19</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Finding dominant structures of nonreversible Markov processes</title>
    <abstract language="eng">Finding metastable sets as dominant structures of Markov processes has been shown to be especially useful in modeling interesting slow dynamics of various real world complex processes. Furthermore, coarse graining of such processes based on their dominant structures leads to better understanding and dimension reduction of observed systems. However, in many cases, e.g. for nonreversible Markov processes, dominant structures are often not formed by metastable sets but by important cycles or mixture of both. This paper aims at understanding and identifying these different types of dominant structures for reversible as well as nonreversible ergodic Markov processes. Our algorithmic approach generalizes spectral based methods for reversible process by using Schur decomposition techniques which can tackle also nonreversible cases. We illustrate the mathematical construction of our new approach by numerical experiments.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-55739</identifier>
    <identifier type="doi">10.1137/15M1032272</identifier>
    <enrichment key="SourceTitle">Appeared in: Multiscale Modeling and Simulation 14(4): 1319-1340</enrichment>
    <author>Natasa Djurdjevac Conrad</author>
    <submitter>Erlinda Koernig</submitter>
    <author>Marcus Weber</author>
    <author>Christof Schütte</author>
    <series>
      <title>ZIB-Report</title>
      <number>15-40</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>nonreversible Markov processes</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>metastable sets</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cycle decomposition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Schur decomposition</value>
    </subject>
    <collection role="msc" number="60J20">Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]</collection>
    <collection role="msc" number="65C40">Computational Markov chains</collection>
    <collection role="msc" number="82C26">Dynamic and nonequilibrium phase transitions (general)</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="persons" number="weber">Weber, Marcus</collection>
    <collection role="projects" number="BMS-Nielsen">BMS-Nielsen</collection>
    <collection role="projects" number="EyeTracking">EyeTracking</collection>
    <collection role="projects" number="NonequiMSM">NonequiMSM</collection>
    <collection role="persons" number="natasa.conrad">Conrad, Natasa</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/5573/ZIB-Report_15-40.pdf</file>
  </doc>
  <doc>
    <id>5116</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2014-08-01</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Modularity of Directed Networks: Cycle Decomposition Approach</title>
    <abstract language="eng">The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarsegrained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits all necessary information about modular structures from the original network.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-51166</identifier>
    <identifier type="doi">10.3934/jcd.2015.2.1</identifier>
    <enrichment key="SourceTitle">Appeared in: Journal of Computational Dynamics 2 (2015) pp. 1-24</enrichment>
    <author>Natasa Djurdjevac Conrad</author>
    <submitter>Erlinda Körnig</submitter>
    <author>Ralf Banisch</author>
    <author>Christof Schütte</author>
    <series>
      <title>ZIB-Report</title>
      <number>14-31</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>directed networks, modules, cycle decomposition, measure of node communication</value>
    </subject>
    <collection role="msc" number="05C81">Random walks on graphs</collection>
    <collection role="msc" number="60J20">Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]</collection>
    <collection role="msc" number="94C15">Applications of graph theory [See also 05Cxx, 68R10]</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="NonequiMSM">NonequiMSM</collection>
    <collection role="persons" number="natasa.conrad">Conrad, Natasa</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/5116/ZIB-Report_14-31.pdf</file>
  </doc>
  <doc>
    <id>4984</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2014-05-07</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Module Detection in Directed Real-World Networks</title>
    <abstract language="eng">We investigate the problem of finding modules (or clusters, communities) in directed networks. Until now, most articles on this topic have been oriented towards finding complete network partitions despite the fact that this often is unwanted. We present a novel random walk based approach for non-complete partitions of the directed network into modules in which some nodes do not belong to only one of the modules but to several or to none at all. The new random walk process is reversible even for directed networks but inherits all necessary information about directions and structure of the original network. We demonstrate the performance of the new method in application to a real-world earthquake network.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-49849</identifier>
    <author>Ralf Banisch</author>
    <submitter>Erlinda Körnig</submitter>
    <author>Christof Schütte</author>
    <author>Natasa Djurdjevac Conrad</author>
    <series>
      <title>ZIB-Report</title>
      <number>14-13</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Module identification and classification</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cycle decomposition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>communication</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>directed networks</value>
    </subject>
    <collection role="msc" number="60J20">Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]</collection>
    <collection role="msc" number="94C15">Applications of graph theory [See also 05Cxx, 68R10]</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="ECMath-CH7">ECMath-CH7</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/4984/ZIB-Report_14-13.pdf</file>
  </doc>
</export-example>
