### 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 eigenvectorbased 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 supracentrality 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 the 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.

Language | English (US) |
---|---|

Pages | 537-574 |

Number of pages | 38 |

Journal | Multiscale Modeling and Simulation |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Eigenvector centrality
- Hubs and authorities
- Multilayer networks
- Ranking systems
- Singular perturbation
- Temporal networks

### ASJC Scopus subject areas

- Chemistry(all)
- Modeling and Simulation
- Ecological Modeling
- Physics and Astronomy(all)
- Computer Science Applications

### Cite this

*Multiscale Modeling and Simulation*,

*15*(1), 537-574. https://doi.org/10.1137/16M1066142

**Eigenvector-based centrality measures for temporal networks.** / Taylor, Dane; Myers, Sean A.; Clauset, Aaron; Porter, Mason A.; Mucha, Peter J.

Research output: Contribution to journal › Article

*Multiscale Modeling and Simulation*, vol. 15, no. 1, pp. 537-574. https://doi.org/10.1137/16M1066142

}

TY - JOUR

T1 - Eigenvector-based centrality measures for temporal networks

AU - Taylor, Dane

AU - Myers, Sean A.

AU - Clauset, Aaron

AU - Porter, Mason A.

AU - Mucha, Peter J.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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 eigenvectorbased 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 supracentrality 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 the 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.

AB - 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 eigenvectorbased 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 supracentrality 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 the 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.

KW - Eigenvector centrality

KW - Hubs and authorities

KW - Multilayer networks

KW - Ranking systems

KW - Singular perturbation

KW - Temporal networks

UR - http://www.scopus.com/inward/record.url?scp=85017001195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017001195&partnerID=8YFLogxK

U2 - 10.1137/16M1066142

DO - 10.1137/16M1066142

M3 - Article

VL - 15

SP - 537

EP - 574

JO - Multiscale Modeling and Simulation

T2 - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 1

ER -