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 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., the divergence rates affect only the edges of either the original or the copy vertex, and (ii) symmetric divergence, i.e., the divergence rates affect with equal probability both the original and the copy vertex. Connected components emerge as the divergence asymmetry rate slightly moves from the complete asymmetric divergence case. Mean-field results of prior published models are nicely reproduced by this generalization. In special cases, the connected component size distribution C_{s} suggests a power-law scaling of the form C_{s}∼s^{-λ} for s>1, e.g., with λ≈5/3 for a divergence rate δ≈0.7.