Degree-Ordered-Percolation on uncorrelated networks

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 $\gamma \le 3$.