Alladi Ramakrishnan Hall

Ricci Forman curvature: From Networks to Hyper-networks and back again

Emil Saucan

Department of Applied Mathematics, ORT Braude College and Technion, Israel

Traditionally, network analysis is based on local properties of vertices, like their degree or clustering coefficient, and their statistical behavior across the network in question, thus concentrating on the elements of the network (nodes), rather than on their interrelations (edges), that define, in effect, the network. We propose an alternative edge-based approach. The geometric tool that enables us to do this is Forman's discretization of Ricci curvature, initially devised for quite general weighted CW complexes. We show that in the limit case of 1-dimensional complexes, i.e. networks (or graphs) this notion is still powerful and expressive enough to allow us to capture not only local, but also global properties of networks, both weighted and unweighted, directed as well as undirected. We show the robustness of this notion and compare it to other, more classical, graph invariants and network descriptors, both on standard model networks and on a variety of real-life networks.
Furthermore, we develop a fitting Ricci flow, and we apply it in the analysis of dynamic networks, and employ it to such tasks as change detection, denoising and clustering of experimental data, as well as to the extrapolation of network evolvement.
Moreover, we consider not only the pairwise correlations in networks, but also the higher order ones, that are especially important in biological and social networks, and apply Forman's original notion to the resulting complexes (hyper-networks) together with an adapted Ricci flow.