Network bipartitioning in the anti-communicability Euclidean space

• Published in: AIMS mathematics Vol. 6; no. 2; pp. 1153 - 1174
• Main Authors: Gómez-Gardeñes, Jesús, Estrada, Ernesto
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: bipartite graphs; communicability; complex networks; network theory
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021070

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.