Emergent networks in fractional percolation

TL;DR

This paper investigates fractional percolation in networks, revealing how partially functional nodes contribute to network resilience, and introduces a coarse-grained bipartite model with power-law distributed supernodes that explains critical phenomena.

Contribution

It develops new equations for fractional percolation, identifies an emergent bipartite structure, and links this to critical exponents near the percolation threshold.

Findings

Percolating cluster includes fully and partially functional nodes.

Emergent bipartite structure with supernodes follows a power-law degree distribution.

The structure explains critical exponents around the percolation threshold.

Abstract

Real networks are vulnerable to random failures and malicious attacks. However, when a node is harmed or damaged, it may remain partially functional, which helps to maintain the overall network structure and functionality. In this paper, we study the network structure for a fractional percolation process [Shang, Phys. Rev. E 89, 012813 (2014)], in which the state of a node can be either fully functional (FF), partially functional (PF), or dysfunctional (D). We develop new equations to calculate the relative size of the percolating cluster of FF and PF nodes, that are in agreement with our stochastic simulations. In addition, we find a regime in which the percolating cluster can be described as a coarse-grained bipartite network, namely, as a set of finite groups of FF nodes connected by PF nodes. Moreover, these groups behave as a set of "supernodes" with a power-law degree distribution. Finally, we show how this emergent structure explains the values of several critical exponents around the percolation threshold.