Ricci curvature of random and empirical directed hypernetworks

Abstract

Relationships in real systems are often not binary, but of a higher order, and therefore cannot be faithfully modelled by graphs, but rather need hypergraphs. In this work, we systematically develop formal tools for analyzing the geometry and the dynamics of hypergraphs. In particular, we show that Ricci curvature concepts, inspired by the corresponding notions of Forman and Ollivier for graphs, are powerful tools for probing the local geometry of hypergraphs. In fact, these two curvature concepts complement each other in the identification of specific connectivity motifs. In order to have a baseline model with which we can compare empirical data, we introduce a random model to generate directed hypergraphs and study properties such as degree of nodes and edge curvature, using numerical simulations. We can then see how our notions of curvature can be used to identify connectivity patterns in the metabolic network of E. coli that clearly deviate from those of our random model. Specifically, by applying hypergraph shuffling to this metabolic network we show that the changes in the wiring of a hypergraph can be detected by Forman Ricci and Ollivier Ricci curvatures.

Computational History: Challenges and Opportunities of Formal Approaches

“…In particular, because of constraints inherited from biological ancestors or cultural traditions, in either case, the resulting structures are highly path dependent. Also, interactions between networks are important, in hierarchical or interdependent ways [68,69] , such as internal regulation and external niche construction for biological species [11] , and social, material, and epistemic or knowledge networks for social structures [56] . In the simplest case, such networks depend on pairwise interactions between elements, and they can then be studied with the well developed tools of graph theory.…”