Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 113-147, January 2021.
Characterizing the importances (i.e., centralities) of nodes in social, biological, and technological networks is a core topic in both network analysis and data science. We present a linear-algebraic framework that generalizes eigenvector-based centralities, including PageRank and hub/authority scores, to provide a common framework for two popular classes of multilayer networks: multiplex networks (which have layers that encode different types of relationships) and temporal networks (in which relationships change over time). Our approach involves the study of joint, marginal, and conditional “supracentralities” that one can calculate from the dominant eigenvector of a supracentrality matrix [Taylor et al., Multiscale Model. Simul., 15 (2017), pp. 537--574; [110] in this paper], which couples centrality matrices that are associated with individual network layers. We extend this prior work (which was restricted to temporal networks with layers that are coupled by adjacent-in-time coupling) by allowing the layers to be coupled through a (possibly asymmetric) interlayer-adjacency matrix $\tilde{\bm{A}}$, where the entry $\tilde{A}_{tt'} \geq 0$ encodes the coupling between layers $t$ and $t'$. Our framework provides a unifying foundation for centrality analysis of multiplex and temporal networks, and it also illustrates a complicated dependency of the supracentralities on the topology and weights of interlayer coupling. By scaling $\tilde{\bm{A}}$ by an interlayer-coupling strength $\omega\ge0$ and developing a singular perturbation theory for the limits of weak ($\omega\to0^+$) and strong ($\omega\to\infty$) coupling, we also reveal an interesting dependence of supracentralities on the right and left dominant eigenvectors of $\tilde{\bm{A}}$. We provide additional theoretical and practical insights by applying our framework to two empirical data sets: a multiplex network of airline transportation in Europe and a temporal network that encodes the graduation and hiring of mathematical scientists at United States universities.

中文翻译：

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多重和时间网络的基于可调谐特征向量的中心性

多尺度建模与仿真，第 19 卷，第 1 期，第 113-147 页，2021 年 1 月。
表征节点在社会、生物和技术网络中的重要性（即中心性）是网络分析和数据科学的核心主题。我们提出了一个线性代数框架，它概括了基于特征向量的中心性，包括 PageRank 和 hub/authority 分数，为两类流行的多层网络提供了一个通用框架：多重网络（具有编码不同类型关系的层）和时间网络（其中关系随时间变化）。我们的方法涉及对联合、边缘和条件“超中心性”的研究，人们可以从超中心性矩阵的主要特征向量[Taylor 等人，多尺度模型。Simul., 15 (2017), pp. 537--574; [110]在本文中]，它耦合了与各个网络层相关联的中心矩阵。我们通过允许层通过（可能是非对称的）层间邻接矩阵 $\tilde{\bm{A }}$，其中条目 $\tilde{A}_{tt'} \geq 0$ 对层 $t$ 和 $t'$ 之间的耦合进行编码。我们的框架为多重和时间网络的中心性分析提供了统一的基础，它还说明了超中心性对层间耦合的拓扑和权重的复杂依赖性。通过通过层间耦合强度 $\omega\ge0$ 缩放 $\tilde{\bm{A}}$ 并开发针对弱 ($\omega\to0^+$) 和强 ($ \omega\to\infty$) 耦合，我们还揭示了超中心性对 $\tilde{\bm{A}}$ 的左右主导特征向量的有趣依赖。通过将我们的框架应用于两个经验数据集，我们提供了额外的理论和实践见解：欧洲航空运输的多元网络和美国大学数学科学家毕业和招聘的时间网络。