Degree-Ordered-Percolation on uncorrelated networks

TL;DR

This paper studies Degree-Ordered Percolation (DOP), where nodes are occupied in degree-descending order, revealing how the percolation threshold and critical properties depend on network heterogeneity, especially in power-law networks.

Contribution

It provides analytical and numerical analysis of DOP on uncorrelated networks, highlighting the dependence of percolation thresholds and critical exponents on degree distribution exponent γ.

Findings

Percolation threshold vanishes for γ ≤ 3 in large networks.

Finite but very small thresholds for 3 < γ < 4, with significant preasymptotic effects.

DOP does not belong to the universality class of random percolation for γ ≤ 3.

Abstract

We analyze the properties of Degree-Ordered Percolation (DOP), a model in which the nodes of a network are occupied in degree-descending order. This rule is the opposite of the much studied degree-ascending protocol, used to investigate resilience of networks under intentional attack, and has received limited attention so far. The interest in DOP is also motivated by its connection with the Susceptible-Infected-Susceptible (SIS) model for epidemic spreading, since a variation of DOP is related to the vanishing of the SIS transition for random power-law degree-distributed networks $P(k) \sim k^{-\gamma}$. By using the generating function formalism, we investigate the behavior of the DOP model on networks with generic value of $\gamma$ and we validate the analytical results by means of numerical simulations. We find that the percolation threshold vanishes in the limit of large networks for $\gamma \le 3$, while it is finite for $\gamma>3$, although its value for $\gamma$ between 3 and 4 is exceedingly small and preasymptotic effects are huge. We also derive the critical properties of the DOP transition, in particular how the exponents depend on the heterogeneity of the network, determining that DOP does not belong to the universality class of random percolation for $\gamma \le 3$.