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 $\hat{I}{´}_c$. The $\hat{I}{´}_c$ found stands near the locus of zero in Euler characteristic for finite-size graphs, known to be indicative of the largest connected component transition. The role of non-interacting vertices in shaping this transition with their presence ($d=0$) and absence ($d=1$) in duplication is also discussed, suggesting a particular transformation of the time variable considered, which yields a singularity locus in the natural logarithm of the absolute value of Euler characteristic in finite-size graphs near to that obtained with $d=1$ but from the model with $d=0$. The findings may suggest implications for bond percolation in these growing graph models.

edit in an inline Eqn. ($\hat{I}½$ rather than $\ddot{I}ˆ$) in Appendix C, caption of Fig.8 (Appendix A), and other minor edits