Biased Percolation on Scale-free Networks

TL;DR

This paper investigates biased edge percolation on scale-free networks, analyzing how degree-dependent edge retention affects network robustness and fragility through analytical and numerical methods.

Contribution

It introduces detailed analysis of biased percolation on scale-free networks, including two reconstruction methods and their critical properties, extending the Fortuin-Kasteleyn framework.

Findings

The theory aligns well with simulation results.

Biased percolation can effectively control network robustness.

Critical exponents are nonuniversal near the transition.

Abstract

Biased (degree-dependent) percolation was recently shown to provide new strategies for turning robust networks fragile and vice versa. Here we present more detailed results for biased edge percolation on scale-free networks. We assume a network in which the probability for an edge between nodes $i$ and $j$ to be retained is proportional to $(k_ik_j)^{-\alpha}$ with $k_i$ and $k_j$ the degrees of the nodes. We discuss two methods of network reconstruction, sequential and simultaneous, and investigate their properties by analytical and numerical means. The system is examined away from the percolation transition, where the size of the giant cluster is obtained, and close to the transition, where nonuniversal critical exponents are extracted using the generating functions method. The theory is found to agree quite well with simulations. By introducing an extension of the Fortuin-Kasteleyn construction, we find that biased percolation is well described by the $q\to 1$ limit of the $q$-state Potts model with inhomogeneous couplings.