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

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 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 reducedparallelly 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.

Emergence of the circle in a statistical model of random cubic graphs

“…Using this notation we have the following expression for the Ollivier curvature in cubic graphs [56]:…”

“…Improvements in the code-partly following insights presented in [56]-have allowed some of these issues to be addressed, and in a future work we aim to present the case of flat surfaces; here, however, we address a more controllable model in which the configuration space consists of cubic (3-regular) graphs. In this model we have a classification of possible classical configurations and strong indications that in the N → ∞ limit, the graphs in question converge to the circle S 1 r of radius r for some fixed r > 0.…”

Emergence of the Circle in a Statistical Model of Random Cubic Graphs

“…• k w denotes the same as k w for w ∈ { u, v } except the shortest cycle is a pentagon instead of a square. Using this notation we have the following expression for the Ollivier curvature in cubic graphs [54]:…”

“…Improvements in the code-partly following insights presented in [54]-have allowed some of these issues to be addressed, and in a future work we aim to present the case of flat surfaces; here, however, we address a more controllable model in which the configuration space consists of cubic (3-regular) graphs. In this model we have a classification of possible classical configurations and strong indications that in the N → ∞ limit, the graphs in question converge to the circle S 1 r of radius r for some fixed r > 0.…”

“…The other essential ingredient is the fact that the system undergoes a continuous phase transition. Since Ricci-flat graphs are likely to exhibit significantly different properties from random regular graphs, and the evidence from[55] suggests that the continuous phase transition was likely to persist as we increase the degree, difficulties in extending this model are thus concentrated on the classification of Ricci-flat graphs; some nontrivial information is contained in[54] but it seems unlikely that we will be able to obtain similarly rigorous results for the case of 4-regular graphs, which in our formalism corresponds to surfaces. Nonetheless there are prospects for similar conclusions to be drawn in the case of surfaces (4-regular graphs) in the form of spectral and Hausdorff dimension results.…”