Largest connected component in duplication-divergence growing graphs with symmetric coupled divergence

Abstract

The largest connected component in duplication-divergence growing graphs with symmetric coupled divergence is studied. Finite-size scaling reveals a phase transition occurring at a divergence rate ${\delta }_c$. The ${\delta }_c$ found is close to the locus of zero in Euler characteristic of finite-size graphs known to reflect the proximity of the largest connected component transition. A close correspondence with the vanishing of a scaling relation exponent for moments of the vertex degree distribution is shown, with such a scaling relation that generalizes a known form for duplication-divergence model graphs. The role of non-interacting vertices in shaping this transition with their presence or absence in duplication is also considered through a particular relation which would result in the two cases being comparable. The findings have relevancy for bond percolation in these growing graph models.

arXiv commentRevised version accepted for publication in Physical Review E with DOI: 10.1103/3zrx-56mx