Growing networks with preferential deletion and addition of edges

Abstract

A preferential attachment model for a growing network incorporating the deletion of edges is studied and the expected asymptotic degree distribution is analyzed. At each time step t = 1 , 2 , … , with probability π 1 > 0 a new vertex with one edge attached to it is added to the network and the edge is connected to an existing vertex chosen proportionally to its degree, with probability π 2 a vertex is chosen proportionally to its degree and an edge is added between this vertex and a randomly chosen other vertex, and with probability π 3 = 1 − π 1 − π 2 < 1 / 2 a vertex is chosen proportionally to its degree and a random edge of this vertex is deleted. The model is intended to capture a situation where high-degree vertices are more dynamic than low-degree vertices in the sense that their connections tend to be changing. A recursion formula is derived for the expected asymptotic fraction p k of vertices with degree k , and solving this recursion reveals that, for π 3 < 1 / 3 , we have p k ∼ k − ( 3 − 7 π 3 ) / ( 1 − 3 π 3 ) , while, for π 3 > 1 / 3 , the fraction p k decays exponentially at rate ( π 1 + π 2 ) / 2 π 3 . There is hence a non-trivial upper bound for how much deletion the network can incorporate without losing the power-law behavior of the degree distribution. The analytical results are supported by simulations.