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Rich-Clubs in Preferential Attachment Networks

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Complex Networks and Their Applications VII (COMPLEX NETWORKS 2018)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 813))

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Abstract

Consider the general random preferential attachment model G(p) for network evolution that allows both node and edge arrivals. Starting with an arbitrary nonempty graph \(G_0\), at each time step, either with probability \(p>0\) a new node arrives and connects to an existing node, or with probability \(1-p\) a new edge is added between two existing nodes. In both cases, the existing nodes are chosen at random with probability proportional to their degree. Letting the \(\delta \) - fraction rich club of the network be the smallest set of nodes which, collectively, hold a \(\delta \) fraction of the total degree in the network, we show that its size is concentrated around \(f_p\left( \delta \right) \cdot n_t\), where \(n_t\) is the number of nodes in the network, and \(f_p\) is a convex continuous piecewise-linear function. This answers the open question of whether or not the \(\delta \) - fraction rich club constitutes a constant fraction of the number of nodes in the network. We provide a full description of \(f_p\). Finally, we compare this with the size of the \(\delta \) - founders of the network defined as the smallest set of the first nodes to enter the network which, collectively, hold a \(\delta \) fraction of the total degree in the network.

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Notes

  1. 1.

    Note that x and \(\mathbf {x'}\) are real-valued vectors and not random variables.

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Author information

Authors and Affiliations

  1. Weizmann Institute of Science, Rehovot, Israel

    Yinon Nahum

Authors
  1. Yinon Nahum
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Correspondence to Yinon Nahum .

Editor information

Editors and Affiliations

  1. Nokia Bell Labs, Cambridge, UK

    Luca Maria Aiello

  2. IUT Lumière, University of Lyon, Bron Cedex, France

    Chantal Cherifi

  3. LE2I UMR CNRS 6306 9, University of Burgundy, Dijon Cedex, France

    Hocine Cherifi

  4. Mathematical Institute, University of Oxford, Oxford, UK

    Renaud Lambiotte

  5. Department of Computer Science and Technology, The Computer Laboratory, University of Cambridge, Cambridge, UK

    Pietro Lió

  6. Center for Complex Networks and Systems Research, School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, USA

    Luis M. Rocha

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Nahum, Y. (2019). Rich-Clubs in Preferential Attachment Networks. In: Aiello, L., Cherifi, C., Cherifi, H., Lambiotte, R., Lió, P., Rocha, L. (eds) Complex Networks and Their Applications VII. COMPLEX NETWORKS 2018. Studies in Computational Intelligence, vol 813. Springer, Cham. https://doi.org/10.1007/978-3-030-05414-4_6

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