- The Robot Crawler Model on Complete k-Partite and Erdős-Rényi Random Graphs(arXiv)
Author : Angus Davidson, Ayalvadi Ganesh
Abstract : Web crawlers are used by internet search engines to gather information about the web graph. In this paper we investigate a simple process which models such software by walking around the vertices of a graph. Once initial random vertex weights have been assigned, the robot crawler traverses the graph deterministically following a greedy algorithm, always visiting the neighbour of least weight and then updating this weight to be the highest overall. We consider the maximum, minimum and average number of steps taken by the crawler to visit every vertex of firstly, complete k-partite graphs and secondly, sparse Erdős-Rényi random graphs. Our work follows on from a paper of Bonato et. al. who introduced the model.
2.A Dynamic Erdős-Rényi Graph Model (arXiv)
Author : Sebastian Rosengren, Pieter Trapman
Abstract : : In this article we introduce a dynamic Erdős-Rényi graph model, in which, independently for each vertex pair, edges appear and disappear according to a Markov on-off process. In studying the dynamic graph we present two main results. The first being on how long it takes for the graph to reach stationarity. We give an explicit expression for this time, as well as proving that this is the fastest time to reach stationarity among all strong stationary times. The second result concerns the time it takes for the dynamic graph to reach a certain number of edges. We give an explicit expression for the expected value of such a time, as well as study its asymptotic behavior. This time is related to the first time the dynamic Erdős-Rényi graph contains a cluster exceeding a certain size
