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The topology of the directed clique complex as a network invariant.

Authors
  • Masulli, Paolo1
  • Villa, Alessandro E P1
  • 1 Neuroheuristic Research Group, University of Lausanne, Internef Quartier UNIL-Dorigny, 1015 Lausanne, Switzerland. , (Switzerland)
Type
Published Article
Journal
SpringerPlus
Publisher
Springer (Biomed Central Ltd.)
Publication Date
Jan 01, 2016
Volume
5
Pages
388–388
Identifiers
DOI: 10.1186/s40064-016-2022-y
PMID: 27047714
Source
Medline
Keywords
License
Unknown

Abstract

We introduce new algebro-topological invariants of directed networks, based on the topological construction of the directed clique complex. The shape of the underlying directed graph is encoded in a way that can be studied mathematically to obtain network invariants such as the Euler characteristic and the Betti numbers. Two different cases illustrate the application of the Euler characteristic. We investigate how the evolution of a Boolean recurrent artificial neural network is influenced by its topology in a dynamics involving pruning and strengthening of the connections, and to show that the topological features of the directed clique complex influence the dynamical evolution of the network. The second application considers the directed clique complex in a broader framework, to define an invariant of directed networks, the network degree invariant, which is constructed by computing the topological invariant on a sequence of sub-networks filtered by the minimum in- or out-degree of the nodes. The application of the Euler characteristic presented here can be extended to any directed network and provides a new method for the assessment of specific functional features associated with the network topology.

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