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

TL;DR

This paper investigates the phase transition of the largest connected component in duplication-divergence growing graphs with symmetric coupled divergence, revealing critical divergence rates and the influence of non-interacting vertices.

Contribution

It introduces a detailed analysis of the phase transition in these graphs, linking divergence rates to the Euler characteristic and the role of non-interacting vertices in the transition.

Findings

Identifies a phase transition at a critical divergence rate near zero in Euler characteristic.

Shows the presence or absence of non-interacting vertices affects the transition.

Suggests implications for bond percolation in these graph models.

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 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.