Comprehensive method for discovering and assessing the rich-club organization in multihypergraphs

Abstract:

The topological structure of the graph-based models constructed for a wide range of the real-world complex systems is characterized by the clear presence of the rich-club organization. Conceptually, such organization implies the tendency of the most important (in accordance to the prescribed metric) vertices to be tightly interconnected with each other and form the cohesive communities referred to as the rich-clubs. Recently, the rich-club ordering has attracted a considerable attention of investigators due to its impact on the robustness and performance of the modeled system as well as the regime of its functioning. At the same time, the prior studies in this direction are entirely limited to the case of simple graphs. This, in turn, points to the existence of the fundamental research gap associated with the need to develop the method for detecting the rich-club organization in multihypergraphs (i.e. hypergraphs allowing the presence of the parallel hyperedges). With a view to bridging the identified gap, this work introduces the family of the original metrics providing the formal way for determining whether the submultihypergraph induced by the most important nodes of multihypergraph could be properly regarded as its rich-club. The proposed metrics are designed to exhaustively capture the complex nature of relationships established in the considered submultihypergraph and, accordingly, take into account not only the number of its hyperedges but also their cardinality and role in ensuring the connectivity of vertices. Furthermore, the paper elaborates the scheme of normalizing the introduced metrics with respect to the reference ensemble of random multihypergraphs possessing the same sequences of vertex degrees and hyperedge cardinalities as the multihypergraph under investigation. Such normalization allows discovering the intentionally emerged rich-club ordering in multihypergraphs that does not follow merely from the structural restrictions imposed by the local properties of vertices and hyperedges. Finally, the paper illustrates the descriptive potential of the developed method by constructing the multihypergraph-based representation of the scientific co-authorship hypernetwork extracted from the IEEE Xplore database and performing the rigorous experimental analysis of its rich-club organization.