Universal spectral properties of multiplex networks allow us to assess the nature of the transition between disease-free and endemic phases in the SIS epidemic spreading model. In a multiplex network, depending on a coupling parameter, $p$, the inverse participation ratio ($IPR$) of the leading eigenvector of the adjacency matrix can be in two different structural regimes: (i) layer-localized and (ii) delocalized. Here we formalize the structural transition point, $p^*$, between these two regimes, showing that there are universal properties regarding both the layer size $n$ and the layer configurations. Namely, we show that $IPR \sim n^{-\delta}$, with $\delta\approx 1$, and revealed an approximately linear relationship between $p^*$ and the difference between the layers' average degrees. Furthermore, we showed that this multiplex structural transition is intrinsically connected with the nature of the SIS phase transition, allowing us to both understand and quantify the phenomenon. As these results are related to the universal properties of the leading eigenvector, we expect that our findings might be relevant to other dynamical processes in complex networks.

中文翻译：

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多重网络中特征向量离域化的普遍性和SIS相变的性质

多重网络的通用光谱特性使我们能够评估 SIS 流行病传播模型中无病和地方病阶段之间过渡的性质。在多路复用网络中，根据耦合参数 $p$，邻接矩阵的前导特征向量的逆参与比 ($IPR$) 可以处于两种不同的结构状态：(i) 层局部化和 (ii)离地。在这里，我们将这两种状态之间的结构转换点 $p^*$ 形式化，表明层大小 $n$ 和层配置都具有通用属性。即，我们展示了 $IPR \sim n^{-\delta}$，$\delta\approx 1$，并揭示了 $p^*$ 与层平均度数之间的差异之间的近似线性关系。此外，我们表明这种多重结构转变与 SIS 相变的性质有着内在的联系，使我们能够理解和量化这一现象。由于这些结果与主要特征向量的普遍性质有关，我们预计我们的发现可能与复杂网络中的其他动态过程有关。