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Connectivity of random graphs after centrality-based vertex removal

Part of: Distribution theory - Probability Stochastic processes Graph theory

Published online by Cambridge University Press:  23 February 2024

Remco van der Hofstad Open the ORCID record for Remco van der Hofstad [Opens in a new window]  and
Manish Pandey Open the ORCID record for Manish Pandey [Opens in a new window]
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
Manish Pandey*
Affiliation:
Eindhoven University of Technology
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands.
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands.
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Abstract

Centrality measures aim to indicate who is important in a network. Various notions of ‘being important’ give rise to different centrality measures. In this paper, we study how important the central vertices are for the connectivity structure of the network, by investigating how the removal of the most central vertices affects the number of connected components and the size of the giant component. We use local convergence techniques to identify the limiting number of connected components for locally converging graphs and centrality measures that depend on the vertex’s neighbourhood. For the size of the giant, we prove a general upper bound. For the matching lower bound, we specialise to the case of degree centrality on one of the most popular models in network science, the configuration model, for which we show that removal of the highest-degree vertices destroys the giant most.

Keywords

Strictly local centrality measurescentrality-based vertex removalnumber of connected componentssize of giantconfiguration model

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 05C80: Random graphs 60E15: Inequalities; stochastic orderings
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 32
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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