Asymptotic coarse Ricci curvature of inhomogeneous random graph

Abstract

In this paper, we study the coarse Ricci curvature of inhomogeneous random graph on vertex set $\left[n\right]≔\{1,2,\dots ,n\}$. In this graph, each pair of vertices $(i,j)$ forms an edge independently with probability $\frac{\lambda }{n^{\alpha }}g\left(\frac{i}{n},\frac{j}{n}\right)$ for some function $g(x,y)\in (0,1]$, constants $\lambda >0$ and $\alpha \in [0,1]$. We derive the asymptotic coarse Ricci curvature of this random graph. Phase transition phenomenon exists as $\alpha$ varies from zero to one.