Focusing of Maximum Vertex Degrees in Random Faulty Scaled Sector Graphs

Abstract

In this paper we study the behavior of maximum out/in-degree of binomial/Poisson random scaled sector graphs in the presence of random vertex and edge faults. We prove that the probability distribution of maximum degrees for random faulty scaled sector graphs with $n$ vertices, where each vertex spans a sector of $\alpha$ radians, with radius $r_n\ll \sqrt{\ln n/n}$, become concentrated on two consecutive integers, under some natural assumptions of faulty probabilities.