Exact Expressions and Reduced Linear Programmes for the Ollivier Curvature in Graphs

TL;DR

This paper investigates the computation of Ollivier curvature in graphs, providing counterexamples to previous claims of exact formulas, proposing reduced linear programs for calculation, and deriving an exact expression for a specific class of graphs.

Contribution

It corrects previous misconceptions, introduces reduced linear programs for efficient computation, and derives an exact formula for a special class of graphs.

Findings

Counterexamples to claims of exact formulas in bipartite and girth 5 graphs.

Reduced linear programs enable parallel computation of Ollivier curvature.

An exact expression is derived for a specific class of graphs with combinatorial constraints.

Abstract

The Ollivier curvature has important applications in discrete geometry and network theory, in particular as a measure of local clustering. The Ollivier curvature is defined in terms of the Wasserstein distance which, in the discrete setting, can be regarded as an optimal solution of a particular linear programme. In certain classes of graph, this linear programme may be solved \textit{a priori} giving rise to exact combinatorial expressions for the Ollivier curvature. It has been claimed by Bhattacharya and Mukherjee (2013) that an exact expression exists for the Ollivier curvature in bipartite graphs and graphs of girth 5; we present counterexamples to these claims and identify the error in the argument of Bhattacharya and Mukherjee. We then repeat the analysis of Bhattacharya and Mukherjee for arbitrary graphs, taking this error into account, and present reduced---parallelly solvable---linear programmes for the calculation of the Ollivier curvature. This allows for potential improvements in the exact numerical evaluation of the Ollivier curvature, though the result heuristically suggests no general exact combinatorial expression for the Ollivier curvature exists. Finally we give an exact expression for the Ollivier curvature in a class of graphs defined by a particular combinatorial constraint motivated by physical considerations.