Laplacian Spectra of Graphs and Cyber-Insurance Protection

Thierry Cohignac (1) , Nabil Kazi-Tani (2)

1 CCR - Caisse centrale de Réassurance
2 SAF - Laboratoire de Sciences Actuarielle et Financière

In this paper, we consider the situation where a given graph has to be protected against communication interruption, through insurance or prevention measures. The goal of the protection buyer is to maintain good connectivity properties of the graph after a malicious attack, giving rise to a virus spread on the network. We model the epidemic spread using the standard Susceptible-Infected-Susceptible (SIS) Markov process. The connectivity of the graph is measured by a function of the average Laplacian spectrum: the second smallest eigenvalue, known as the algebraic connectivity. Using standard results on eigenvalues optimization, we recast the algebraic connectivity maximization as a semidefinite optimization problem, for which a solution exists and can be efficiently numerically computed. Our results allow to hierarchize the edges of a graph, giving more importance to some edges for which the protection demand is high, hence making optimal insurance demand directly depend on the underlying network topology.

Mots clés: Algebraic connectivity; Large graphs; Susceptible-Infected-Susceptible; Cyber Insurance

Domaines: Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Thierry Cohignac, Nabil Kazi-Tani. Laplacian Spectra of Graphs and Cyber-Insurance Protection. 2020. ⟨hal-02911859⟩