A combinatorial Yamabe flow in three dimensions

A combinatorial version of Yamabe flow is presented based on Euclidean triangulations coming from sphere packings. The evolution of curvature is then derived and shown to satisfy a heat equation. The Laplacian in the heat equation is shown to be a geometric analogue of the Laplacian of Riemannian geometry, although the maximum principle need not hold. It is then shown that if the flow is nonsingular, the flow converges to a constant curvature metric.Comment: 20 pages, 5 figures. The paper arxiv:math.MG/0211195 was absorbed into its new version and this pape

Forman-Ricci flow for change detection in large dynamic data sets

We present a viable solution to the challenging question of change detection in complex networks inferred from large dynamic data sets. Building on Forman's discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. Furthermore we adapt the classical Ricci flow that already proved to be a powerful tool in image processing and graphics, to the case of undirected and weighted networks. The application of the proposed method on peer-to-peer networks yields insights into topological properties and the structure of their underlying data.Comment: Conference paper, accepted at ICICS 2016. (Updated version

Irreducibility of Markov Chains on simplicial complexes, the Spectrum of the Discrete Hodge Laplacian and Homology

Random walks on graphs are a fundamental concept in graph theory and play a crucial role in solving a wide range of theoretical and applied problems in discrete math, probability, theoretical computer science, network science, and machine learning. The connection between Markov chains on graphs and their geometric and topological structures is the main reason why such a wide range of theoretical and practical applications exist. Graph connectedness ensures irreducibility of a Markov chain. The convergence rate to the stationary distribution is determined by the spectrum of the graph Laplacian which is associated with lower bounds on graph curvature. Furthermore, walks on graphs are used to infer structural properties of underlying manifolds in data analysis and manifold learning. However, an important question remains: can similar connections be established between Markov chains on simplicial complexes and the topology, geometry, and spectral properties of complexes? Additionally, can we gain topological, geometric, or analytic information about a manifold by defining appropriate Markov chains on its triangulations? These questions are not only theoretically important but answers to them provide powerful tools for the analysis of complex networks that go beyond the analysis of pairwise interactions. In this paper, we provide an integrated overview of the existing results on random walks on simplicial complexes, using the novel perspective of signed graphs. This perspective sheds light on previously unknown aspects such as irreducibility conditions. We show that while up-walks on higher dimensional simplexes can never be irreducible, the down walks become irreducible if and only if the complex is orientable. We believe that this new integrated perspective can be extended beyond discrete structures and enables exploration of classical problems for triangulable manifolds.Comment: 16 pages, 3 figure