Research article Special Issues

Network bipartitioning in the anti-communicability Euclidean space

  • Received: 14 January 2020 Accepted: 11 June 2020 Published: 11 November 2020
  • MSC : 05C82, 05C50, 05C69, 68R10

  • We define the anti-communicability function for the nodes of a simple graph as the nondiagonal entries of exp (-A). We prove that it induces an embedding of the nodes into a Euclidean space. The anti-communicability angle is then defined as the angle spanned by the position vectors of the corresponding nodes in the anti-communicability Euclidean space. We prove analytically that in a given k-partite graph, the anti-communicability angle is larger than 90° for every pair of nodes in different partitions and smaller than 90° for those in the same partition. This angle is then used as a similarity metric to detect the "best" k-partitions in networks where certain level of edge frustration exists. We apply this method to detect the "best" k-partitions in 15 real-world networks, finding partitions with a very low level of "edge frustration". Most of these partitions correspond to bipartitions but tri- and pentapartite structures of real-world networks are also reported.

    Citation: Jesús Gómez-Gardeñes, Ernesto Estrada. Network bipartitioning in the anti-communicability Euclidean space[J]. AIMS Mathematics, 2021, 6(2): 1153-1174. doi: 10.3934/math.2021070

    Related Papers:

  • We define the anti-communicability function for the nodes of a simple graph as the nondiagonal entries of exp (-A). We prove that it induces an embedding of the nodes into a Euclidean space. The anti-communicability angle is then defined as the angle spanned by the position vectors of the corresponding nodes in the anti-communicability Euclidean space. We prove analytically that in a given k-partite graph, the anti-communicability angle is larger than 90° for every pair of nodes in different partitions and smaller than 90° for those in the same partition. This angle is then used as a similarity metric to detect the "best" k-partitions in networks where certain level of edge frustration exists. We apply this method to detect the "best" k-partitions in 15 real-world networks, finding partitions with a very low level of "edge frustration". Most of these partitions correspond to bipartitions but tri- and pentapartite structures of real-world networks are also reported.


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