标题： 用于多层网络的多项式特征值方法 显示英文标题

标题： A polynomial eigenvalue approach for multiplex networks

摘要： 我们探讨了多层网络矩阵表示的块结构，引入了一种新的形式化方法，以处理其谱特性作为层间耦合参数的函数。 这种方法使我们能够基于传统特征值问题的解释得出有趣的结果。 更具体地说，我们降低了矩阵的维度，但增加了特征多项式的威力，即一个多项式特征值问题。 这种方法乍看之下可能显得反直觉，但它使我们能够将2层多层系统的二次问题与聚合网络的谱联系起来，并推导出谱的界限，以及许多其他有趣的分析见解。 此外，它还允许直接获得关于层间耦合下特征值行为的分析和数值见解。 我们的研究包括超邻接矩阵、超拉普拉斯矩阵和概率转移矩阵，这使我们能够从多层网络的结构相变角度来审视我们的结果。 我们认为，这种形式化方法和所报告的结果将使未来能够为多层网络推导出新的结果。 显示英文摘要

摘要： We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting results based on an interpretation of the traditional eigenvalue problem. More specifically, we reduce the dimensionality of our matrices but increase the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. Such an approach may sound counterintuitive at first glance, but it allows us to relate the quadratic problem for a 2-Layer multiplex system with the spectra of the aggregated network and to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits to directly obtain analytical and numerical insights on the eigenvalue behavior as a function of the coupling between layers. Our study includes the supra-adjacency, supra-Laplacian, and the probability transition matrices, which enable us to put our results under the perspective of structural phases in multiplex networks. We believe that this formalism and the results reported will make it possible to derive new results for multiplex networks in the future.