## Abstract

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.

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## Notes

Note that \(\delta (t)\) or \(\delta (\mathsf {k})\), and \(\delta (G)\) are commonly used as standard notations for the disagreement vector in consensus theory and for the minimum degree of graph \(G\) in graph theory, respectively. Our intent was to follow standard notations in the respective fields; therefore, to avoid ambiguities further in the paper, \(\delta (t)\) or \(\delta (\mathsf {k})\) will always be used for the disagreement vector, whereas \(\delta (G)\) will always denote the minimum degree of graph \(G\).

For \(n=12\) there exists one more graph with \(2n-6\) edges with this property [7].

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## Acknowledgments

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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Veremyev, A., Boginski, V. & Pasiliao, E.L. Analytical characterizations of some classes of optimal strongly attack-tolerant networks and their Laplacian spectra.
*J Glob Optim* **61**, 109–138 (2015). https://doi.org/10.1007/s10898-014-0141-y

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DOI: https://doi.org/10.1007/s10898-014-0141-y