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Graph Metrics for Temporal Networks

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Temporal Networks

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

Temporal networks, i.e., networks in which the interactions among a set of elementary units change over time, can be modelled in terms of time-varying graphs, which are time-ordered sequences of graphs over a set of nodes. In such graphs, the concepts of node adjacency and reachability crucially depend on the exact temporal ordering of the links. Consequently, all the concepts and metrics proposed and used for the characterisation of static complex networks have to be redefined or appropriately extended to time-varying graphs, in order to take into account the effects of time ordering on causality. In this chapter we discuss how to represent temporal networks and we review the definitions of walks, paths, connectedness and connected components valid for graphs in which the links fluctuate over time. We then focus on temporal node–node distance, and we discuss how to characterise link persistence and the temporal small-world behaviour in this class of networks. Finally, we discuss the extension of classic centrality measures, including closeness, betweenness and spectral centrality, to the case of time-varying graphs, and we review the work on temporal motifs analysis and the definition of modularity for temporal graphs.

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Notes

  1. 1.

    In order to avoid confusion with the size Δt of the time-window used to define the temporal snapshot of a time-varying graph, here we preferred to use Δτ instead of the original Δt proposed by the authors of [20]. Also, notice that the authors use to call events what we have called here contacts.

  2. 2.

    It is possible to prove that a random walk on a graph always converges towards a stationary state, independently of the initial condition, if the adjacency matrix of the graph is primitive, which is the case for the vast majority of real graphs.

  3. 3.

    The interested reader can find the derivation of (23) in [27] and in the Supplemental Information of the same paper.

  4. 4.

    This is required to ensure the existence of a stationary state for the Laplacian dynamics on the graph.

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Acknowledgements

This work was funded in part through EPSRC Project MOLTEN (EP/I017321/1).

Author information

Authors and Affiliations

  1. Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, Cambridge, CB3 0FD, UK

    Vincenzo Nicosia, John Tang & Cecilia Mascolo

  2. Laboratorio sui Sistemi Complessi, Scuola Superiore di Catania, Via Valdisavoia 9, 95123, Catania, Italy

    Vincenzo Nicosia & Vito Latora

  3. School of Computer Science, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    Mirco Musolesi

  4. Dipartimento di Matematica e Informatica, Universitá di Catania, Via S. Sofia 64, 95123, Catania, Italy

    Giovanni Russo

  5. School of Mathematical Sciences, Queen Mary, University of London, E1 4NS, London, UK

    Vito Latora

  6. Dipartimento di Fisica e Astronomia and INFN, Universitá di Catania, Via S. Sofia 64, 95123, Catania, Italy

    Vito Latora

Authors
  1. Vincenzo Nicosia
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  2. John Tang
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  3. Cecilia Mascolo
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  4. Mirco Musolesi
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  5. Giovanni Russo
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  6. Vito Latora
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Corresponding authors

Correspondence to Vincenzo Nicosia or Mirco Musolesi .

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Editors and Affiliations

  1. Ice Lab, Department of Physics, Umeå University, Ice Lab, 3rd floor, NA wing, Naturvetarhuset, Umeå, 901 87, Sweden

    Petter Holme

  2. and Computational Sciences (BECS), Department of Biomedical Engineering, Aalto University, Rakentajanaukio, Otaniemi campus 2, Aalto, 00076, Finland

    Jari Saramäki

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Nicosia, V., Tang, J., Mascolo, C., Musolesi, M., Russo, G., Latora, V. (2013). Graph Metrics for Temporal Networks. In: Holme, P., Saramäki, J. (eds) Temporal Networks. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36461-7_2

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  • DOI: https://doi.org/10.1007/978-3-642-36461-7_2

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