Eigenvector-Based Centrality Measures for Temporal Networks

Abstract

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 NT × NT 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.

Asymmetry through time dependency

“…The parameter is included to account for the fact that the identity matrices represent "between layer" connections (generally linking node i at time k to node i at times k − 1 and k + 1) which are inherently different to the "within layer" weights arising from the network data. The key idea in [27] is to apply a standard static network centrality algorithm to the supra-centrality matrix (13 in the Katz-based setting), the supra-centrality matrix M is symmetric. It then becomes apparent that any static network algorithm applied to M must be oblivious to the type of temporal asymmetry that we have discussed.…”

Temporal Network Analysis of Literary Texts

“…To do this we combine the theoretical framework of Woloch [30] and Moretti [19,20,21] for literary studies with techniques developed by Taylor et al [26] for temporal networks on the mathematical side. To the best of our knowledge the present paper is the first one where these methods are applied to literature.…”

“…This paper is organized as follows: In the first section we briefly review the theoretical framework of character networks in literature while addressing the first question: how to choose a network. This is followed by an exposition of the method propounded in [26] In this sense a story really is a complex system. This is a matter-of-course fact in literary studies but here lies the link to the application of network theory to narratives: taking networks as the natural theoretical framework for studying the interdependence between agents that act through space and time, the main task one faces is how to disregard, in a principled way, aspects of narratives which might not render themselves amenable to a mathematical treatment.…”

“…The way they introduced dynamics is by looking at some relevant measures chapter by chapter and comparing them. Taylor et al devised in [26] a way of generalizing eigenvector-based measures usually studied in time-independent scenarios to temporal networks. These authors look at snapshots of the network at different time layers and treating inter-layer connections (same time) and intra-layer connections (different times) as being essentially different.…”

“…One may then use all the results from spectral analysis already known to work for aggregate networks. This ideas were tested with 3 networks: the exchange of PhD's in mathematics in the United States, the costarring networks of top-billed actors during the Golden Age of Hollywood and citations of decisions of the American Supreme Court [26]. The results presented in [1] represent a valuable approach towards incorporating some sort of dynamics into networks as well as in the way they attribute values to edges, but we believe that the methods of Taylor and coworkers capture the dynamics in a more consistent and rigorous way.…”