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The temporal rich club phenomenon

Abstract

Identifying the hidden organizational principles and relevant structures of complex networks is fundamental to understand their properties. To this end, uncovering the structures involving the prominent nodes in a network is an effective approach. In temporal networks, the simultaneity of connections is crucial for temporally stable structures to arise. Here, we propose a measure to quantitatively investigate the tendency of well-connected nodes to form simultaneous and stable structures in a temporal network. We refer to this tendency as the temporal rich club phenomenon, characterized by a coefficient defined as the maximal value of the density of links between nodes with a minimal required degree, which remain stable for a certain duration. We illustrate the use of this concept by analysing diverse data sets and their temporal properties, from the role of cohesive structures in relation to processes unfolding on top of the network to the study of specific moments of interest in the evolution of the network.

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Fig. 1: The TRC concept.
Fig. 2: US air transportation temporal network.
Fig. 3: Primary school temporal network.
Fig. 4: Temporal network of information-sharing neurons.

Data availability

All the data used in this work are available at https://github.com/nicolaPedre/Temporal-Rich-Club/.

Code availability

The analysis presented here were performed in Python. Example notebooks are available at https://github.com/nicolaPedre/Temporal-Rich-Club/.

The randomized reference models were obtained thanks to the publicly available notebooks by the authors of ref. 34, namely at https://github.com/mgenois/RandTempNet.

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Acknowledgements

N.P. has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement no. 713750. Also, the project has been carried out with the financial support of the Regional Council of Provence-Alpes-Côte d’Azur and with the financial support of A*MIDEX (ANR-11-IDEX-0001-02), funded by the Investissements d’Avenir project funded by the French Government, managed by the French National Research Agency (ANR).

D.B. is supported by the European Union Innovative Training Network “i- CONN” (H2020 ITN 859937).

A.B. is supported by Agence Nationale de la Recherche (ANR) project DATAREDUX (ANR-19-CE46-0008).

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N.P., D.B. and A.B. designed the study and the experiment. N.P. developed the numerical tools, and produced the figures. N.P., D.B. and A.B. analysed the data and results and wrote and reviewed the manuscript.

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Correspondence to Alain Barrat.

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Nature Physics thanks Shi Zhou, Dan Braha and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Notes 1–5.

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Pedreschi, N., Battaglia, D. & Barrat, A. The temporal rich club phenomenon. Nat. Phys. (2022). https://doi.org/10.1038/s41567-022-01634-8

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