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 $\gamma \le 3$.

中文翻译：

---

不相关网络上的度序渗透

我们分析了度序渗透 (DOP) 的特性，DOP 是一种网络节点按度降序占用的模型。这条规则与被广泛研究的程度递增协议相反，用于研究网络在故意攻击下的弹性，迄今为止受到的关注有限。对 DOP 的兴趣还源于它与流行病传播的易感-感染-易感 (SIS) 模型的联系，因为 DOP 的变化与随机幂律度分布网络的 SIS 过渡的消失有关 $P (k) \sim k^{-\gamma}$。通过使用生成函数形式，我们研究了 DOP 模型在具有 $\gamma$ 泛型值的网络上的行为，并通过数值模拟验证了分析结果。我们发现，对于 $\gamma\le 3$，渗流阈值在大型网络的极限内消失，而 $\gamma>3$ 时它是有限的，尽管 $\gamma$ 在 3 和 4 之间的值非常小并且渐近前效应是巨大的。我们还推导出 DOP 转换的关键特性，特别是指数如何依赖于网络的异质性，确定 DOP 不属于 $\gamma\le 3$ 的随机渗透的普遍性类别。