Issue 
RAIROOper. Res.
Volume 51, Number 4, OctoberDecember 2017



Page(s)  1055  1076  
DOI  https://doi.org/10.1051/ro/2017008  
Published online  24 November 2017 
The vertex attack tolerance of complex networks
Southern Illinois University, Edwardsville, Illinois, USA.
gercal@siue.edu
Received: 19 May 2016
Accepted: 27 January 2017
The purpose of this work is fourfold: (1) We propose a new measure of network resilience in the face of targeted node attacks, vertex attack tolerance, represented mathematically as $\mathit{\tau}\mathrm{\left(}\mathit{G}\mathrm{\right)}\mathrm{=}{\mathrm{min}}_{\mathit{S}\mathrm{\subset}\mathit{V}}\frac{\mathrm{\left}\mathit{S}\mathrm{\right}}{\mathrm{}\mathit{V}\mathrm{}\mathit{S}\mathrm{}{\mathit{C}}_{\mathrm{max}}\mathrm{(}\mathit{V}\mathrm{}\mathit{S}\mathrm{\left)}\mathrm{\right}\mathrm{+}\mathrm{1}}$, and prove that for dregular graphs τ(G) = Θ(Φ(G)) where Φ(G) denotes conductance, yielding spectral bounds as corollaries. (2) We systematically compare τ(G) to known resilience notions, including integrity, tenacity, and toughness, and evidence the dominant applicability of τ for arbitrary degree graphs. (3) We explore the computability of τ, first by establishing the hardness of approximating unsmoothened vertex attack tolerance $\mathit{\tau \u0302}\mathrm{\left(}\mathit{G}\mathrm{\right)}\mathrm{=}{\mathrm{min}}_{\mathit{S}\mathrm{\subset}\mathit{V}}\frac{\mathrm{\left}\mathit{S}\mathrm{\right}}{\mathrm{}\mathit{V}\mathrm{}\mathit{S}\mathrm{}{\mathit{C}}_{\mathrm{max}}\mathrm{(}\mathit{V}\mathrm{}\mathit{S}\mathrm{\left)}\mathrm{\right}}$ under various plausible computational complexity assumptions, and then by presenting empirical results on the performance of a betweenness centrality based heuristic algorithm applied not only to τ but several other hard resilience measures as well. (4) Applying our algorithm, we find that the random scalefree network model is more resilient than the Barabasi−Albert preferential attachment model, with respect to all resilience measures considered.
Mathematics Subject Classification: 68R10 / 90B15 / 05C50 / 68Q25
Key words: Graph theory / resilience / ScaleFree networks / spectral Gap / approximation Hardness / Heuristic Algorithms
© EDP Sciences, ROADEF, SMAI 2017
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