Block Matrix Formulations for Evolving Networks

Fenu, Caterina; Higham, Desmond J.

doi:10.1137/16m1076988

Cited by 22 publications

(21 citation statements)

References 32 publications

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“…We have studied inter-layer edges between nearestneighbor node-layer pairs; that is, a node-layer pair (i, t) is adjacent to (i, t − 1) and (i, t + 1), so the edges form an undirected chain that bridges physical node i across the T time layers. Other strategies should be explored (see discussions in [72]), including ones with inter-layer edges that are directed (e.g., other temporal generalizations of communicability centrality [38]). Directed inter-layer edges are able to represent causal scenarios, but such causal coupling can yield (supra-centrality) matrices that are not irreducible, which is a problematic situation for eigenvector-based centrality measures.…”

Section: Citations Of United States Supreme Court Decisions (Scd)mentioning

confidence: 99%

“…One common feature of existing generalizations of centralities for temporal networks is that they illustrate the importance of studying an entire temporal network, as opposed to aggregating temporal layers into a single (time-independent) network or analyzing the time layers in isolation from one another [60,61]. Specifically, studying a layer-aggregated network prevents one from studying centrality trajectories (i.e., how centrality changes over time), and studying the time layers in isolation does not account for the temporal orderings of edges, which can be crucial for determining centralities in a temporal setting [33,38,50,51,84]. To provide additional context, we highlight that dynamical processes can behave vastly differently on temporal versus layer-aggregated networks.…”

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confidence: 99%

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Eigenvector-Based Centrality Measures for Temporal Networks

Taylor¹,

Myers²,

Clauset³

et al. 2017

Multiscale Model. Simul.

173

198

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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.

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Section: Citations Of United States Supreme Court Decisions (Scd)mentioning

confidence: 99%

mentioning

confidence: 99%

Eigenvector-Based Centrality Measures for Temporal Networks

Taylor¹,

Myers²,

Clauset³

et al. 2017

Multiscale Model. Simul.

173

198

View full text Add to dashboard Cite

show abstract

“…Moreover, their variant relies on parameters defining what is to be considered as relevant path lengths and path duration, which complicates the analysis. Fenu et al [26] explore the different methods for representing a dynamic network as a block matrix where each column/row corresponds to a pair composed of a node and a time instant. They propose a method to construct such a block matrix that allows to compute dynamic centrality metrics in an efficient way.…”

Section: Related Workmentioning

confidence: 99%

“…This further shows that both methods produce different results when the activity is low. This is probably due to the observation made by Fenu et al [26] that Temporal Eigenvector considers paths that do not respect time and can therefore go backwards in time. This explains why nodes that are permanently inactive at the end of the dataset can have a non-zero rank.…”

Section: Global Observationsmentioning

confidence: 99%

Centrality Metrics in Dynamic Networks: A Comparison Study

Ghanem

Magnien

Tarissan

2019

IEEE Trans. Netw. Sci. Eng.

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For a long time, researchers have worked on defining different metrics able to characterize the importance of nodes in static networks. Recently, researchers have introduced extensions that consider the dynamics of networks. These extensions study the time-evolution of the importance of nodes, which is an important question that has yet received little attention in the context of temporal networks. They follow different approaches for evaluating a node's importance at a given time and the value of each approach remains difficult to assess. In order to study this question more in depth, we compare in this paper a method we recently introduced to three other existing methods. We use several datasets of different nature, and show and explain how these methods capture different notions of importance.We also show that in some cases it might be meaningless to try to identify nodes that are globally important. Finally, we highlight the role of inactive nodes, that still can be important as a relay for future communications.

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“…In our particular case, we measure the vitality with respect to the eigenvector centralities of nodes. Recently, Fenu and Higham [5] have argued that the way the supra-centrality matrix is constructed in eq. (1) can be problematic when one considers directed graphs.…”

Section: Temporal Network: Some Definitionsmentioning

confidence: 99%

Temporal Network Analysis of Literary Texts

Prado

Dahmen

Bazzan

et al. 2016

Advs. Complex Syst.

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We study temporal networks of characters in literature focusing on Alice's Adventures in Wonderland (1865) by Lewis Carroll and the anonymous La Chanson de Roland (around 1100). The former, one of the most influential pieces of nonsense literature ever written, describes the adventures of Alice in a fantasy world with logic plays interspersed along the narrative. The latter, a song of heroic deeds, depicts the Battle of Roncevaux in 778 A.D. during Charlemagne's campaign on the Iberian Peninsula. We apply methods recently developed by Taylor et al. [26] to find time-averaged eigenvector centralities, Freeman indices and vitalities of characters. We show that temporal networks are more appropriate than static ones for studying stories, as they capture features that the time-independent approaches fail to yield.

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Block Matrix Formulations for Evolving Networks

Cited by 22 publications

References 32 publications

Eigenvector-Based Centrality Measures for Temporal Networks

Eigenvector-Based Centrality Measures for Temporal Networks

Centrality Metrics in Dynamic Networks: A Comparison Study

Temporal Network Analysis of Literary Texts

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