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Stream graphs and link streams for the modeling of interactions over time

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Abstract

Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the intrinsically temporal and structural nature of interactions, which calls for a dedicated formalism. In this paper, we generalize graph concepts to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher level objects such as quotient graphs, line graphs, k-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex cases.

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Notes

  1. Unless explicitly specified, we always consider simple and undirected graphs and stream graphs; we discuss more general cases in Sect. 20.

  2. A partition of a set X into k parts is a family \((P_1, P_2, \ldots , P_k)\) of k subsets of X, such that \(\cup _i P_i = X\) and \(P_i \cap P_j = \emptyset\) for all \(i\ne j\).

  3. This is a generalization to stream graphs of the \(\Delta\)-density introduced in Viard and Latapy (2014) for link streams.

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Acknowledgements

This work is funded in part by the European Commission H2020 FETPROACT 2016-2017 program under Grant 732942 (ODYCCEUS), by the ANR (French National Agency of Research) under Grants ANR-15-CE38-0001 (AlgoDiv) and ANR-13-CORD-0017-01 (CODDDE), by the French program “PIA - Usages, services et contenus innovants” under Grant O18062-44430 (REQUEST), and by the Ile-de-France program FUI21 under Grant 16010629 (iTRAC). We warmly thank the many colleagues and friends who read preliminary versions of this work and provided invaluable feedback.

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  1. Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7606, LIP6, 75005, Paris, France

    Matthieu Latapy, Tiphaine Viard & Clémence Magnien

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  1. Matthieu Latapy
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  2. Tiphaine Viard
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Correspondence to Matthieu Latapy.

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Latapy, M., Viard, T. & Magnien, C. Stream graphs and link streams for the modeling of interactions over time. Soc. Netw. Anal. Min. 8, 61 (2018). https://doi.org/10.1007/s13278-018-0537-7

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