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

Degree Sequence of Random Permutation Graphs

In this paper, we study the asymptotics of the degree sequence of permutation graphs associated with a sequence of random permutations. The limiting finite-dimensional distributions of the degree proportions are established using results from graph and permutation limit theories. In particular, we show that for a uniform random permutation, the joint distribution of the degree proportions of the vertices labeled ⌈nr1⌉,⌈nr2⌉,…,⌈nrs⌉ in the associated permutation graph converges to independent random variables D(r1), D(r2),…, D(rs), where D(ri)∼Unif(ri,1−ri), for ri ∈ [0,1] and i ∈ {1,2,…,s}. Moreover, the degree proportion of the mid-vertex (the vertex labeled n/2) has a central limit theorem, and the minimum degree converges to a Rayleigh distribution after an appropriate scaling. Finally, the asymptotic finite-dimensional distributions of the permutation graph associated with a Mallows random permutation is determined, and interesting phase transitions are observed. Our results extend to other nonuniform measures on permutations as well

Application-based COVID-19 micro-mobility solution for safe and smart navigation in pandemics

Short distance travel and commute being inevitable, safe route planning in pandemics for micro-mobility, i.e., cycling and walking, is extremely important for the safety of oneself and others. Hence, we propose an application-based solution using COVID-19 occurrence data and a multi-criteria route planning technique for cyclists and pedestrians. This study aims at objectively determining the routes based on various criteria on COVID-19 safety of a given route while keeping the user away from potential COVID-19 transmission spots. The vulnerable spots include places such as a hospital or medical zones, contained residential areas, and roads with a high connectivity and influx of people. The proposed algorithm returns a multi-criteria route modeled on COVID-19-modified parameters of micro-mobility and betweenness centrality considering COVID-19 avoidance as well as the shortest available safe route for user ease and shortened time of outside environment exposure. We verified our routing algorithm in a part of Delhi, India, by visualizing containment zones and medical establishments. The results with COVID-19 data analysis and route planning suggest a safer route in the context of the coronavirus outbreak as compared to normal navigation and on average route extension is within 8%–12%. Moreover, for further advancement and post-COVID-19 era, we discuss the need for adding open data policy and the spatial system architecture for data usage, as a part of a pandemic strategy. The study contributes new micro-mobility parameters adapted for COVID-19 and policy guidelines based on aggregated contact tracing data analysis maintaining privacy, security, and anonymity