A Simple Differential Geometry for Networks and its Generalizations

  title={A Simple Differential Geometry for Networks and its Generalizations},
  author={Emil Saucan and Areejit Samal and Jurgen Jost},
  booktitle={COMPLEX NETWORKS},
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… 

A simple differential geometry for complex networks

Abstract We introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in

Integral geometric Hopf conjectures

The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf

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

A systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of (hyper)edges, instead of vertices is developed, which utilizes so-called network curvatures.

Network geometry and market instability

Examining the daily returns from a set of stocks comprising the USA S&P-500 and the Japanese Nikkei-225 over a 32-year period, and monitoring the changes in the edge-centric network curvatures finds that the different geometric measures capture well the system-level features of the market and hence can distinguish between the normal or ‘business-as-usual’ periods and all the major market crashes.

Network-centric Indicators for Fragility in Global Financial Indices

Over the last 2 decades, financial systems have been studied and analyzed from the perspective of complex networks, where the nodes and edges in the network represent the various financial components

Geometric Sampling of Networks

This work makes appeal to three types of discrete curvature, namely the graph Forman-, full Forman- and Haantjes-Ricci curvatures for edge-based and node-based sampling, and considers fitting Ricci flows and employ them for the detection of networks' backbone.



Curvature of Hypergraphs via Multi-Marginal Optimal Transport

Empirical experiments demonstrate that coarse scalar curvatures detects “bridges” across connected components in hypergraphs, akin to the behavior of coarse Ricci curvatures on graphs.

Metric Curvatures and their Applications 2: Metric Ricci Curvature and Flow

This part of the overview of the different metric curvatures and their various applications focuses on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and proposes another approach to the metrization of RicCI curvature, based on Forman's discretization.

Metric curvatures and their applications I

We present, in a natural, developmental manner, the main types of metric curvatures and investigate their relationship with the notions of Hausdorff and Gromov-Hausdorff distances, which by now have

Forman's Ricci curvature - From networks to hypernetworks

It is shown that, in fact, a geometric unifying approach is possible to hypernetworks, by viewing them as polyhedral complexes endowed with a simple, yet, the powerful notion of curvature - the Forman Ricci curvature.

Ricci curvature of Markov chains on metric spaces

Coarse geometry of evolving networks

This paper investigates edge-based properties, and the Euler characteristic of a network as a global characteristic is defined directly directly, using a construction of Bloch which yields a discrete Gauß-Bonnet theorem.

Intrinsic Metrics on Graphs: A Survey

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed,

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Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature

  • R. Forman
  • Mathematics
    Discret. Comput. Geom.
  • 2003
A combinatorial analogue of Bochner's theorems is derived, which demonstrates that there are topological restrictions to a space having a cell decomposition with everywhere positive curvature.