Termination of Multipartite Graph Series Arising from Complex Network Modelling

TL;DR

This paper introduces a novel multipartite graph series model for complex networks that overcomes previous limitations by ensuring termination and computational tractability, capturing clique overlaps effectively.

Contribution

It proposes a restricted factorisation process for multipartite graphs that always terminates, enabling better modeling of complex network clique overlaps.

Findings

The process terminates for any graph.

The resulting multipartite graph has notable combinatorial properties.

The model is computationally feasible for complex networks.

Abstract

An intense activity is nowadays devoted to the definition of models capturing the properties of complex networks. Among the most promising approaches, it has been proposed to model these graphs via their clique incidence bipartite graphs. However, this approach has, until now, severe limitations resulting from its incapacity to reproduce a key property of this object: the overlapping nature of cliques in complex networks. In order to get rid of these limitations we propose to encode the structure of clique overlaps in a network thanks to a process consisting in iteratively factorising the maximal bicliques between the upper level and the other levels of a multipartite graph. We show that the most natural definition of this factorising process leads to infinite series for some instances. Our main result is to design a restriction of this process that terminates for any arbitrary graph. Moreover, we show that the resulting multipartite graph has remarkable combinatorial properties and is closely related to another fundamental combinatorial object. Finally, we show that, in practice, this multipartite graph is computationally tractable and has a size that makes it suitable for complex network modelling.