Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T16:03:56.632Z Has data issue: false hasContentIssue false
  • English
  • Français

Depths in hooking networks

Published online by Cambridge University Press:  11 May 2021

Colin Desmarais  and
Hosam Mahmoud Open the ORCID record for Hosam Mahmoud [Opens in a new window]
Colin Desmarais
Affiliation:
Department of Mathematics, Uppsala University, Uppsala, Sweden. E-mail: colin.desmarais@math.uu.se
Hosam Mahmoud
Affiliation:
Department of Statistics, The George Washington University, Washington, DC 20052, USA. E-mail: hosam@gwu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A hooking network is built by stringing together components randomly chosen from a set of building blocks (graphs with hooks). The vertices are endowed with “affinities” which dictate the attachment mechanism. We study the distance from the master hook to a node in the network chosen according to its affinity after many steps of growth. Such a distance is commonly called the depth of the chosen node. We present an exact average result and a rather general central limit theorem for the depth. The affinity model covers a wide range of attachment mechanisms, such as uniform attachment and preferential attachment, among others. Naturally, the limiting normal distribution is parametrized by the structure of the building blocks and their probabilities. We also take the point of view of a visitor uninformed about the affinity mechanism by which the network is built. To explore the network, such a visitor chooses the nodes uniformly at random. We show that the distance distribution under such a uniform choice is similar to the one under random choice according to affinities.

Keywords

Distance in graphLimit lawNetworkPreferential attachmentRandom graphSmall world
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 941 - 949
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albert, R., Jeong, H., & Barabási, A. (1999). Diameter of the world-wide web. Nature 401: 130–131.CrossRefGoogle Scholar
Barabási, A. (2018). Network science. Cambridge, UK: Cambridge University Press.Google Scholar
Desmarais, C. & Holmgren, C. (2020). Normal limit laws for vertex degrees in randomly grown hooking networks and bipolar networks. Electronic Journal of Combinatorics 27: P2.45.CrossRefGoogle Scholar
Devroye, L. (1988). Applications of the theory of records in the study of random trees. Acta Informatica 26: 123–130.CrossRefGoogle Scholar
Gopaladesikan, M., Mahmoud, H., & Ward, M.D. (2014). Building random trees from blocks. Probability in the Engineering and Informational Sciences 28: 67–81.CrossRefGoogle Scholar
Gut, A. (2013). Probability: a graduate course, 2nd ed. New York: Springer.CrossRefGoogle Scholar
Kleinberg, J. (2000). Navigation in a small world. Nature 406: 845.CrossRefGoogle Scholar
Mahmoud, H. (1991). Limiting distributions for path lengths in recursive trees. Probability in the Engineering and Informational Sciences 5: 53–59.CrossRefGoogle Scholar
Mahmoud, H. (1992). Distances in random plane-oriented recursive trees. Journal of Computational and Applied Mathematics 4: 237–245.Google Scholar
Mahmoud, H. (2019). Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment. Advances in Applied Mathematics 111: 101930.CrossRefGoogle Scholar
Watts, D. & Strogatz, S. (1998). Collective dynamics of small-world networks. Nature 393: 440–442.CrossRefGoogle Scholar