Abstract
Characterizing topological properties and anomalous behaviors of higher-dimensional topological spaces via notions of curvatures is by now quite common in mainstream physics and mathematics, and it is therefore natural to try to extend these notions from the non-network domains in a suitable way to the network science domain. In this article we discuss one such extension, namely Ollivier’s discretization of Ricci curvature. We first motivate, define and illustrate the Ollivier–Ricci Curvature. In the next section we provide some “not-previously-published” bounds on the exact and approximate computation of the curvature measure. In the penultimate section we review a method based on the linear sketching technique for efficient approximate computation of the Ollivier–Ricci network curvature. Finally in the last section we provide concluding remarks with pointers for further reading.
Highlights
It is quite common in mainstream physics and mathematics [1,2] to characterize topological properties and anomalous behaviors of higher-dimensional topological spaces via notions of curvatures of these spaces, e.g., in general relativity, extreme variations of four dimensional space-time curvatures via geodesic incompleteness lead to characterizations of black-holes [3]
It is natural to try to extend these notions from the non-network domains e.g., from continuous metric spaces or from higher-dimensional geometric objects) in a suitable way to the network science domain so that non-trivial new topological characteristics of networks can be captured
In this paper we have reviewed some computational aspects of the Ollivier–Ricci curvature for networks, and shown a few simple computational bounds
Summary
It is quite common in mainstream physics and mathematics [1,2] to characterize topological properties and anomalous behaviors of higher-dimensional topological spaces via notions of (local and global) curvatures of these spaces, e.g., in general relativity, extreme variations of four dimensional space-time curvatures via geodesic incompleteness lead to characterizations of black-holes [3]. The topological association itself can be carried out several ways This type of curvature originated relatively recently, there are already a number of papers investigating properties of these measures and applying them to real-world networks, e.g., see [8,15,16,17,18]. The network curvature discussed in this paper is another discretization of Ricci curvature, namely Ollivier’s discretization [19,20,21,22], dubbed as the “Ollivier–Ricci curvature” Both Ollivier–Ricci curvature and Forman-Ricci curvature assign measures that assign a number to each edge of the given network, but the numbers are calculated in quite different ways in these two curvatures since they capture different metric properties of a Riemannian manifold. DistG (u, v) (or dist(u, v)) denote the distance (i.e., number of edges in a shortest path) between the nodes u and v in G
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