Journal of Global Optimization

, Volume 61, Issue 1, pp 109–138 | Cite as

Analytical characterizations of some classes of optimal strongly attack-tolerant networks and their Laplacian spectra

  • Alexander Veremyev
  • Vladimir BoginskiEmail author
  • Eduardo L. Pasiliao


This paper analytically characterizes certain classes of low-diameter strongly attack-tolerant networks of arbitrary size, which are globally optimal in the sense that they contain the minimum possible number of edges. Strong attack tolerance property of level \(R\) implies that a network preserves connectivity and diameter after the deletion of up to \(R-1\) network elements (vertices and/or edges). In addition to identifying such optimal network configurations, we explicitly derive their entire Laplacian spectra, that is, all eigenvalues and eigenvectors of the graph Laplacian matrix. Each of these eigenvalues is by itself a solution to a global optimization problem; thus, the results of this study show that these optimization problems yield analytical solutions for the considered classes of networks. As an important special case, we show that the algebraic connectivity (i.e., the second-smallest eigenvalue of the Laplacian) considered as a function on all networks with fixed vertex connectivity \(R\) reaches its maximum on the optimal \(R\)-robust 2-club, which has diameter 2 and strong attack tolerance of level \(R\). We also demonstrate that the obtained results have direct implications on the exact calculation of convergence speed of consensus algorithms utilizing the entire Laplacian spectrum, which is in contrast to traditionally used simulation-based estimates through just the algebraic connectivity.


Optimal attack-tolerant network topologies Laplacian spectra  Algebraic connectivity 



This research was supported in part by the Air Force Research Laboratory (AFRL) and the Defense Threat Reduction Agency (DTRA). This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. The research was performed while the first author held a National Research Council Research Associateship Award at AFRL.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Alexander Veremyev
    • 1
  • Vladimir Boginski
    • 2
    Email author
  • Eduardo L. Pasiliao
    • 1
  1. 1.Munitions DirectorateAir Force Research LaboratoryEglin AFBUSA
  2. 2.Defense-Oriented Operations Research Laboratory, Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

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