A Simple Differential Geometry for Networks and Its Generalizations

Abstract

Based on two classical notions of curvature for curves in general metric spaces, namely the Menger and Haantjes curvatures, we introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts. These new types of curvature, that apply to weighted and unweighted, directed or undirected networks, are far more intuitive and easier to compute, than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature, and derived via a fitting discrete Gauss-Bonnet Theorem, are quite flexible. We also propose even simpler and more intuitive substitutes of the Haantjes curvature, that allow for even faster and easier computations in large-scale networks.

Network geometry and market instability

“…Among notions of metric, and indeed, discrete curvature, Menger [ 67 ] has proposed the simplest and earliest definition whereby he defines the curvature of metric triangles T formed by three points in the space as the reciprocal 1/ R ( T ) of the radius R ( T ) of the circumscribed circle of a triangle T . Recently, some of us [ 55 , 68 ] have adapted Menger’s definition to networks. Let ( M , d ) be a metric space and T = T ( a , b , c ) be a triangle with sides a , b , c , then the Menger curvature of T is given by where p = ( a + b + c )/2.…”

“…Furthermore, it is clear from the above formula that Menger curvature is always positive. Following the differential geometric approach, the Menger–Ricci (MR) curvature of an edge e in a network can be defined as [ 55 , 68 ] where T e ∼ e denote the triangles adjacent to the edge e . Intuitively, if an edge is part of several triangles in the network, such an edge will have high positive MR curvature ( figure 1 ).…”

“…In networks, can be replaced by a path π = v 0 , v 1 , …, v n between two nodes v 0 and v n , and the subtending chord by the edge e = ( v 0 , v n ) between the two nodes. Recently, some of us [ 55 , 68 ] have defined the Haantjes curvature of such a simple path π as where, if the graph is a metric graph, l ( v 0 , v n ) = d ( v 0 , v n ), that is the shortest path distance between nodes v 0 and v n . In particular, for the combinatorial metric (or unweighted graphs), we obtain that , where π = v 0 , v 1 , …, v n is as above.…”

“…Note that considering simple paths in graphs concords with the classical definition of Haantjes curvature, since a metric arc is, by its very definition, a simple curve. Thereafter, the Haantjes–Ricci (HR) curvature of an edge e [ 55 , 68 ] can be defined as where π ∼ e denote the paths that connect the nodes anchoring the edge e . Note that while MR curvature considers only triangles or simple paths of length 2 between two nodes anchoring an edge in unweighted graphs, the HR curvature considers even longer paths between the same two nodes anchoring an edge ( figure 1 ).…”

Edge-based analysis of networks: curvatures of graphs and hypergraphs

“…In this situation, we and our collaborators have developed the research paradigm of a relation based analysis of networks (for instance Sreejith et al 2016;Weber et al 2017;Sreejith et al 2017;Samal et al 2018;Saucan et al 2019;Painter et al 2019;Saucan et al 2020;Leal et al 2020;Farzam et al 2020. That is, we evaluate relations and associate measures to them whose statistics across the network then can provide structural insight.…”