Exact and Asymptotic Results on Coarse Ricci Curvature of Graphs

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we obtain the exact formulas for Ollivier’s Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature that hold for a wide class of graphs, and extend earlier results where the Ricci-curvature for graphs with girth 6 was obtained. We also prove a general lower bound on the Ricci-curvature in terms of the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize the Ricci-flat graphs of girth 5. Moreover, using our general lower bound and the Birkhoff–von Neumann theorem, we give the first necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci-curvature of random bipartite graphs G (n, n, p) and random graphs G (n, p), in various regimes of p

Ricci curvature on polyhedral surfaces via optimal transportation

The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu & Yau, Jost & Liu have used and extended this notion for graphs giving estimates for the curvature and hence the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific but crucial case of polyhedral surfaces

Self-Assembly of Geometric Space from Random Graphs

We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice

Information geometric methods for complexity

Research on the use of information geometry (IG) in modern physics has witnessed significant advances recently. In this review article, we report on the utilization of IG methods to define measures of complexity in both classical and, whenever available, quantum physical settings. A paradigmatic example of a dramatic change in complexity is given by phase transitions (PTs). Hence we review both global and local aspects of PTs described in terms of the scalar curvature of the parameter manifold and the components of the metric tensor, respectively. We also report on the behavior of geodesic paths on the parameter manifold used to gain insight into the dynamics of PTs. Going further, we survey measures of complexity arising in the geometric framework. In particular, we quantify complexity of networks in terms of the Riemannian volume of the parameter space of a statistical manifold associated with a given network. We are also concerned with complexity measures that account for the interactions of a given number of parts of a system that cannot be described in terms of a smaller number of parts of the system. Finally, we investigate complexity measures of entropic motion on curved statistical manifolds that arise from a probabilistic description of physical systems in the presence of limited information. The Kullback-Leibler divergence, the distance to an exponential family and volumes of curved parameter manifolds, are examples of essential IG notions exploited in our discussion of complexity. We conclude by discussing strengths, limits, and possible future applications of IG methods to the physics of complexity.Comment: review article, 60 pages, no figure