Some asymptotic properties of duplication graphs

TL;DR

This paper investigates the asymptotic properties of duplication graphs, revealing that they often develop power-law degree distributions and retain structural remnants of initial configurations, especially in duplication-dominated regimes.

Contribution

It introduces and analyzes three models of duplication graph growth, highlighting the emergence of power-law distributions and the influence of initial conditions on asymptotic behavior.

Findings

Power-law degree distributions emerge in all models.

Duplication-dominated regimes show slow convergence to asymptotic distributions.

Initial network structure influences long-term degree distribution properties.

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 non-zero 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.