INHOMOGENEOUS RANDOM GRAPHS, ISOLATED VERTICES, AND POISSON APPROXIMATION

作者：Penrose Mathew D^*

来源：Journal of Applied Probability, 2018, 55(1): 112-136.

DOI：10.1017/jpr.2018.9

• Inhomogeneous random graph
• random connection model
• stochastic block model
• latent variable model
• random geometric graph
• Poisson approximation
• Stein's method
• U-statistic

Consider a graph on randomly scattered points in an arbitrary space, with any two points x, y connected with probability phi(x, y). Suppose the number of points is large but the mean number of isolated points is O(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

• 出版日期2018-3