9 research outputs found
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
Diffusion in multi-dimensional solids using Forman's combinatorial differential forms
The formulation of combinatorial differential forms, proposed by Forman for
analysis of topological properties of discrete complexes, is extended by
defining the operators required for analysis of physical processes dependent on
scalar variables. The resulting description is intrinsic, different from the
approach known as Discrete Exterior Calculus, because it does not assume the
existence of smooth vector fields and forms extrinsic to the discrete complex.
In addition, the proposed formulation provides a significant new modelling
capability: physical processes may be set to operate differently on cells with
different dimensions within a complex. An application of the new method to the
heat/diffusion equation is presented to demonstrate how it captures the effect
of changing properties of microstructural elements on the macroscopic behavior.
The proposed method is applicable to a range of physical problems, including
heat, mass and charge diffusion, and flow through porous media
Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets
We present a viable geometric solution for the detection of dynamic effects in complex networks. 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. We study the classical Ricci-flow in a network-theoretic setting and introduce an analytic tool for characterizing dynamic effects. The formalism suggests a computational method for change detection and the identification of fast evolving network regions and yields insights into topological properties and the structure of the underlying data