A Markov Chain Approach to RandomlyGrown Graphs

Abstract

A Markov chain approach to the study of randomly grown graphs isproposed and applied to some popular models that have found use in biologyand elsewhere. For most randomly grown graphs used in biology,it is not known whether the graph or properties of the graph converge (insome sense) as the number of vertices becomes large. Particularly, we studythe behaviour of the degree sequence, that is, the number of vertices withdegree $0,1,\dots ,$in large graphs, and apply our results to the partial duplicationmodel. We further illustrate the results by application to real data.