Eigenvector-Based Centrality Measures for Temporal Networks (1507.01266v3)

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 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.

• The paper introduces a generalization of eigenvector centrality tailored for analyzing networks that change over time, filling a gap in temporal network theory.
• It develops a supra-centrality matrix framework that couples temporal layers and uses a singular perturbation approach for efficient computation of time-averaged and "mover" centrality scores.
• Empirical studies on academic, historical actor, and legal citation networks demonstrate the method's effectiveness in revealing both persistent influence and dynamic shifts in node importance over time.

Eigenvector-Based Centrality Measures for Temporal Networks

The paper "Eigenvector-Based Centrality Measures for Temporal Networks" introduces a comprehensive generalization of eigenvector centrality for networks whose structure evolves over time. Traditional centrality measures are predominantly tailored for static, or time-independent, networks, but this work extends them to accommodate networks with temporal dynamics. This extension is significant, as temporal networks are increasingly prevalent across various domains, from social dynamics to biological systems.

The core of this approach is the development of a supra-centrality matrix that aggregates centrality measures across time layers. Each layer represents a snapshot of the network at a given time, and the centrality of nodes is then analyzed in a coupled temporal framework. The paper considers networks with $N$ nodes and represents them across $T$ time layers, constituting a $NT \times NT$ supra-centrality matrix. The nodes' centrality for each time layer is influenced by its relationships in adjacent layers, facilitated by inter-layer coupling with a parameter $\omega$, which adjusts the coupling strength between layers.

In temporal networks, it is crucial to consider not only the intra-layer dynamics but also inter-layer temporal consistency. This paper takes such a nuanced approach by defining the joint centrality, reflecting the significance of nodes considering their persistence through time. The concepts of marginal and conditional centralities further refine this approach, allowing for decomposition and analysis of centrality trajectories over time. Such decompositions aid in understanding the temporal evolution of node importance without losing the context provided by temporal interdependencies.

The literature has examples of temporal extensions for measures like closeness and betweenness centrality; however, eigenvector-based centrality measures have been less explored in the context of temporal network theory. This paper fills that gap by making eigenvector-based measures adaptable for temporal networks, ensuring that they capture time-varying dynamics effectively.

A mathematical novelty introduced in the paper involves a singular perturbation approach applied in the strong-coupling regime ($\omega \to \infty$), which derives expressions for time-averaged centralities and explores first-order changes in centrality scores, termed "first-order-mover" scores. Such derivations allow for computational efficiency as they require only the solution of linear equations of size $N \times N$, avoiding $NT \times NT$ computations. Consequently, time-averaged eigenvector centralities provide insights into nodes' overall importance, whereas mover scores capture changes and dynamic shifts, which are crucial for understanding the transitional phases within networks.

The empirical explorations with three datasets—mathematical Ph.D. exchange networks in the U.S., networks of top-billed actors in Hollywood's Golden Age, and the citation network of the U.S. Supreme Court decisions—demonstrate the method's applicability. These case studies underscore the method's prowess in revealing both stable and dynamically fluctuating nodes' roles over time. For instance, in academic and legal networks, time-averaged authorities provided a measure of sustained influence, while mover scores identified periods of rapid change and emerging importance in historical timelines.

In conclusion, the paper extends the operational utility of eigenvector-based methods in network analysis by providing tools that respect the intrinsic temporal structures of evolving networks. The supra-centrality matrix framework and its associated mathematical formalism afford new insights into the paper of complex systems, potentially unlocking better strategies for data-driven decision-making in dynamically changing environments. Future applications of this approach could deepen understanding across a range of temporal networks in fields like epidemiology, information spreading in social media, and temporal biochemical interactions in systems biology.