BKT nature of the percolation transition in duplication–divergence graphs with mutation

Determine whether the infinite-order percolation transition observed in the duplication–divergence growing graph model with mutation—where each new vertex duplicates a randomly chosen vertex, retains each duplicate edge with probability p=1−δ, independently adds edges to non-neighbors at a Poisson rate β, and is considered in the regime δ=1 (p=0) with β>0—belongs to the Berezinskii–Kosterlitz–Thouless universality class. Specifically, characterize the critical behavior and scaling (e.g., order parameter, susceptibility, correlation length), and provide a renormalization-group or equivalent theoretical description that establishes or refutes the BKT nature of the transition in this non-equilibrium random graph model.

Background

In the reviewed duplication–divergence model with mutation, when β>0 and δ=1, the size distribution of connected components displays distinct behaviors across percolating and non-percolating phases, and the overall transition is of infinite order. This regime can be cast within a broader class of growing-network processes where a new vertex attaches to k existing vertices with probability p_k, recovering known universal properties.

These features have been noted as reminiscent of the Berezinskii–Kosterlitz–Thouless (BKT) transition in statistical physics. However, the precise correspondence—if any—between the graph percolation transition and BKT criticality remains unresolved, motivating a rigorous analysis of universality and scaling in the duplication–divergence-with-mutation setting.