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The Directed Age-Dependent Random Connection Model with Arc Reciprocity

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Modelling and Mining Networks (WAW 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14671))

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Abstract

We introduce a directed spatial random graph model aimed at modelling certain aspects of social media networks. We provide two variants of the model: an infinite version and an increasing sequence of finite graphs that locally converge to the infinite model. Both variants have in common that each vertex is placed into Euclidean space and carries a birth time. Given locations and birth times of two vertices, an arc is formed from younger to older vertex with a probability depending on both birth times and the spatial distance of the vertices. If such an arc is formed, a reverse arc is formed with probability depending on the ratio of the endpoints’ birth times. Aside from the local limit result connecting the models, we investigate degree distributions, two different clustering metrics and directed percolation.

CM’s research is supported by the German Research Foundation (DFG) Priority Programme 2265, grant no. 4439160. LL gratefully received support by the Leibniz Association within the Leibniz Junior Research Group on Probabilistic Methods for Dynamic Communication Networks as part of the Leibniz Competition (grant no. J105/2020).

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Notes

  1. 1.

    One example that comes to mind are scientific citation networks in which a publication sends an arc to every publication it references.

  2. 2.

    Clearly, it suffices to update the root vertex only at the birth times of new vertices, since we are merely interested in the one dimensional marginal distributions of the resulting family of rooted graphs.

  3. 3.

    In fact, \(D_*\) is the quotient space of equivalence classes of rooted graphs up to graph isomorphism, but we do not distinguish between a graph and its isomorphism class here. For more background on \(D_*\) and local weak convergence see [1].

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Lukas Lüchtrath

  2. Johannes Gutenberg Universität Mainz, Mainz, Germany

    Christian Mönch

Authors
  1. Lukas Lüchtrath
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  2. Christian Mönch
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Correspondence to Lukas Lüchtrath .

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Editors and Affiliations

  1. Tutte Institute for Mathematics and Computing, Ottawa, ON, Canada

    Megan Dewar

  2. SGH Warsaw School of Economics, Warsaw, Poland

    Bogumił Kamiński

  3. SGH Warsaw School of Economics, Warsaw, Poland

    Daniel Kaszyński

  4. SGH Warsaw School of Economics, Warsaw, Poland

    Łukasz Kraiński

  5. Toronto Metropolitan University, Toronto, ON, Canada

    Paweł Prałat

  6. Tutte Institute for Mathematics and Computing, Ottawa, ON, Canada

    François Théberge

  7. SGH Warsaw School of Economics, Warsaw, Poland

    Małgorzata Wrzosek

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Lüchtrath, L., Mönch, C. (2024). The Directed Age-Dependent Random Connection Model with Arc Reciprocity. In: Dewar, M., et al. Modelling and Mining Networks. WAW 2024. Lecture Notes in Computer Science, vol 14671. Springer, Cham. https://doi.org/10.1007/978-3-031-59205-8_7

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  • DOI: https://doi.org/10.1007/978-3-031-59205-8_7

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