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Random Walks on Stochastic Temporal Networks

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Temporal Networks

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

Abstract

In the study of dynamical processes on networks, there has been intense focus on network structure—i.e., the arrangement of edges and their associated weights—but the effects of the temporal patterns of edges remains poorly understood. In this chapter, we develop a mathematical framework for random walks on temporal networks using an approach that provides a compromise between abstract but unrealistic models and data-driven but non-mathematical approaches. To do this, we introduce a stochastic model for temporal networks in which we summarize the temporal and structural organization of a system using a matrix of waiting-time distributions. We show that random walks on stochastic temporal networks can be described exactly by an integro-differential master equation and derive an analytical expression for its asymptotic steady state. We also discuss how our work might be useful to help build centrality measures for temporal networks.

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Notes

  1. 1.

    This condition holds in our setting, as we have assumed that the underlying graph \(\mathcal{G}\) of potential edges is strongly connected.

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Acknowledgements

This chapter is based on [20], which contains additional calculations and numerical simulations. RL would like to acknowledge support from FNRS (MIS-2012-F.4527.12) and Belspo (PAI Dysco). MAP acknowledges a research award (#220020177) from the James S. McDonnell Foundation and a grant from the EPSRC (EP/J001759/1). We thank T. Carletti, J.-C. Delvenne, P. J. Mucha, M. Rosvall, and J. Saramäki for fruitful discussions, and we thank P. Holme for helpful comments in his review of this manuscript.

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Authors and Affiliations

  1. Department of Physics, University of Oxford, Oxford, OX1 3RH, UK

    Till Hoffmann

  2. Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute and CABDyN Complexity Centre, University of Oxford, Oxford, OX1 3LB, UK

    Mason A. Porter

  3. Department of Mathematics, University of Namur, Namur, Belgium

    Renaud Lambiotte

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  1. Till Hoffmann
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  2. Mason A. Porter
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  3. Renaud Lambiotte
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Correspondence to Renaud Lambiotte .

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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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Hoffmann, T., Porter, M.A., Lambiotte, R. (2013). Random Walks on Stochastic 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_15

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  • DOI: https://doi.org/10.1007/978-3-642-36461-7_15

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