title: Some asymptotic properties of duplication graphs.
abstract: Duplication graphs are graphs that grow by duplication of existing vertices, and are important models of biological networks, including protein-protein interaction networks and gene regulatory networks. Three models of graph growth are studied: pure duplication growth, and two two-parameter models in which duplication forms one element of the growth dynamics. A power-law degree distribution is found to emerge in all three models. However, the parameter space of the latter two models is characterized by a range of parameter values for which duplication is the predominant mechanism of graph growth. For parameter values that lie in this "duplication-dominated" regime, it is shown that the degree distribution either approaches zero asymptotically, or approaches a nonzero power-law degree distribution very slowly. In either case, the approach to the true asymptotic degree distribution is characterized by a dependence of the scaling exponent on properties of the initial degree distribution. It is therefore conjectured that duplication-dominated, scale-free networks may contain identifiable remnants of their early structure. This feature is inherited from the idealized model of pure duplication growth, for which the exact finite-size degree distribution is found and its asymptotic properties studied.
authors.affiliation: Keck Graduate Institute of Applied Life Sciences, 535 Watson Drive, Claremont, California 91711, USA. araval@kgi.edu
authors.forename: Alpan
authors.initials: A
authors.lastname: Raval
mesh_terms: D015233:Models, Statistical; D009154:Mutation; D016571:Neural Networks, Computer; D011336:Probability