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

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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$R$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>R</mi> </math> </EquationSource> </InlineEquation> implies that a network preserves connectivity and diameter after the deletion of up to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$R-1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>R</mi> <mo>-</mo> <mn>1</mn> </mrow> </math> </EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$R$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>R</mi> </math> </EquationSource> </InlineEquation> reaches its maximum on the optimal <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$R$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>R</mi> </math> </EquationSource> </InlineEquation>-robust 2-club, which has diameter 2 and strong attack tolerance of level <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$R$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>R</mi> </math> </EquationSource> </InlineEquation>. 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. Copyright Springer Science+Business Media New York 2015