PDE and Applied Mathematics Seminar[ Edit ]
Moderator: Nir Gavish
In recent years a significant process of ``geometrization of Combinatorics" has occurred and flourished. In particular, the of the role of curvature - and mainly of Ricci curvature - in the study of Complex Networks has been recognized. Traditionally, network analysis is based on local properties of vertices, like their degree or clustering coefficient, rather than on their interrelations (edges), that define, in effect, the network. Ricci curvature, by its very definition, allows an alternative, edge-based approach. We focus on Forman's discrete 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 (multiplex networks, hypernetworks) together with an adapted Ricci flow. We also show that this higher dimensional approach naturally renders a Persistent Homology scheme for Complex Networks.We conclude with more - in part quite fashionable - applications.
We consider Laplace eigenfunctions of a metric graph satisfying Neumann-Kirchhoff conditions on every vertex. The nodal count of a given eigenfunction is the number of points at which it vanishes. The nodal count of the n-th eigenfunction was shown to be bounded between n-1 and n-1+\beta, where \beta if the first Betti number of the graph. The difference between the nodal count and n-1 is called the nodal surplus. Berkolaiko et al. showed that the n-th nodal surplus equals to a magnetic stability index of the n-th eigenvalue.
We present recent results on the statistics of the nodal surplus including new results and progress on the conjectured universal behavior for large graphs.
This talk is based on joint works with Ram Band (Technion) and Gregory Berkolaiko (Texas A&M).