Divergence asymmetry and connected components in a general duplication-divergence graph model

Abstract

This Letter introduces a generalization of known duplication-divergence models for growing random graphs. This general duplication-divergence model includes a new coupled divergence asymmetry rate, which allows to obtain the structure of random growing networks by duplication-divergence in a continuous range of configurations between two known limit cases (i) complete asymmetric divergence, i.e., divergence rates affect only edges of either the original or the copy vertex, and (ii) symmetric divergence, i.e., divergence rates affect equiprobably both the original and the copy vertex. Multiple connected sub-graphs (of order greater than one) (of order greater than one) emerge as the divergence asymmetry rate slightly moves from the complete asymmetric divergence case. Mean-field results of priorly published models are nicely reproduced by this generalization. In special cases, the connected components size distribution $C_s$ suggests a power-law scaling of the form $C_s \sim s^{-\lambda}$ for $s>1$, e.g., with $\lambda \approx 5/3$ for divergence rate $\delta \approx 0.7$.