How Many Vertices Does a Random Walk Miss in a Network with Moderately Increasing the Number of Vertices?

TL;DR

This paper analyzes how a random walk covers vertices in a dynamic graph model where the number of vertices increases moderately, revealing that almost all vertices are covered asymptotically.

Contribution

It introduces a new model of dynamic graphs with increasing vertices and analyzes the cover time of random walks in this setting, which is a novel contribution.

Findings

Random walks cover all but a constant number of vertices asymptotically.

The model captures dynamic networks with increasing vertices.

Analysis provides insights into cover times in evolving networks.

Abstract

Real networks are often dynamic. In response to it, analyses of algorithms on {\em dynamic networks} attract more and more attentions in network science and engineering. Random walks on dynamic graphs also have been investigated actively in more than a decade, where in most cases the edge set changes but the vertex set is static. The vertex sets are also dynamic in many real networks. Motivated by a new technology of the analysis of random walks on dynamic graphs, this paper introduces a simple model of graphs with increasing the number of vertices, and presents an analysis of random walks associated with the cover time on such graphs. In particular, we reveal that a random walk asymptotically covers the vertices all but a constant number if the vertex set grows {\em moderately}.