Eigenvector-Based Centrality Measures for Temporal Networks

Numerous centrality measures have been developed to quantify the importances of nodes in time-independent networks, and many of them can be expressed as the leading eigenvector of some matrix. With the increasing availability of network data that changes in time, it is important to extend such eigenvector-based centrality measures to time-dependent networks. In this paper, we introduce a principled generalization of network centrality measures that is valid for any eigenvector-based centrality. We consider a temporal network with N nodes as a sequence of T layers that describe the network during different time windows, and we couple centrality matrices for the layers into a supra-centrality matrix of size NTxNT whose dominant eigenvector gives the centrality of each node i at each time t. We refer to this eigenvector and its components as a joint centrality, as it reflects the importances of both the node i and the time layer t. We also introduce the concepts of marginal and conditional centralities, which facilitate the study of centrality trajectories over time. We find that the strength of coupling between layers is important for determining multiscale properties of centrality, such as localization phenomena and the time scale of centrality changes. In the strong-coupling regime, we derive expressions for time-averaged centralities, which are given by the zeroth-order terms of a singular perturbation expansion. We also study first-order terms to obtain first-order-mover scores, which concisely describe the magnitude of nodes' centrality changes over time. As examples, we apply our method to three empirical temporal networks: the United States Ph.D. exchange in mathematics, costarring relationships among top-billed actors during the Golden Age of Hollywood, and citations of decisions from the United States Supreme Court.Comment: 38 pages, 7 figures, and 5 table

Understanding Complex Systems: From Networks to Optimal Higher-Order Models

To better understand the structure and function of complex systems, researchers often represent direct interactions between components in complex systems with networks, assuming that indirect influence between distant components can be modelled by paths. Such network models assume that actual paths are memoryless. That is, the way a path continues as it passes through a node does not depend on where it came from. Recent studies of data on actual paths in complex systems question this assumption and instead indicate that memory in paths does have considerable impact on central methods in network science. A growing research community working with so-called higher-order network models addresses this issue, seeking to take advantage of information that conventional network representations disregard. Here we summarise the progress in this area and outline remaining challenges calling for more research.Comment: 8 pages, 4 figure

Higher-Order Aggregate Networks in the Analysis of Temporal Networks: Path structures and centralities

Recent research on temporal networks has highlighted the limitations of a static network perspective for our understanding of complex systems with dynamic topologies. In particular, recent works have shown that i) the specific order in which links occur in real-world temporal networks affects causality structures and thus the evolution of dynamical processes, and ii) higher-order aggregate representations of temporal networks can be used to analytically study the effect of these order correlations on dynamical processes. In this article we analyze the effect of order correlations on path-based centrality measures in real-world temporal networks. Analyzing temporal equivalents of betweenness, closeness and reach centrality in six empirical temporal networks, we first show that an analysis of the commonly used static, time-aggregated representation can give misleading results about the actual importance of nodes. We further study higher-order time-aggregated networks, a recently proposed generalization of the commonly applied static, time-aggregated representation of temporal networks. Here, we particularly define path-based centrality measures based on second-order aggregate networks, empirically validating that node centralities calculated in this way better capture the true temporal centralities of nodes than node centralities calculated based on the commonly used static (first-order) representation. Apart from providing a simple and practical method for the approximation of path-based centralities in temporal networks, our results highlight interesting perspectives for the use of higher-order aggregate networks in the analysis of time-stamped network data.Comment: 27 pages, 13 figures, 3 table