Decreasing the spectral radius of a graph by link removals

Abstract

The decrease of the spectral radius, an important characterizer of network dynamics, by removing links is investigated. The minimization of the spectral radius by removing m links is shown to be an NP-complete problem, which suggests considering heuristic strategies. Several greedy strategies are compared, and several bounds on the decrease of the spectral radius are derived. The strategy that removes that link l = i ∼ j with largest product (x 1 ) i (x 1 ) j of the components of the eigenvector x 1 belonging to the largest adjacency eigenvalue is shown to be superior to other strategies in most cases. Furthermore, a scaling law where the decrease in spectral radius is inversely proportional to the number of nodes N in the graph is deduced. Another sublinear scaling law of the decrease in spectral radius versus the number m of removed links is conjectured.

Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated

“…An important consequence of this independence assumption and the inequalities (1, 2) is that NIMFA always upper bounds the probability of infection of each node in the network and hence lower bounds the epidemic threshold. From a practical point of view, when designing [13] or controlling [14,15] a network against the epidemics by using NIMFA, the upper bound property ensures that the network is safeguarded from long-term, massive infection.…”

Epidemics in networks with nodal self-infection and the epidemic threshold

“…If the effective spreading rate τ = β δ > τ c , the virus persists and a nonzero fraction of the nodes are infected, whereas for τ τ c , the epidemic dies out and the network is virus free in the steady state. From the point of view of network protection against viral infections, the epidemic threshold τ c is the key parameter in the design of immunization strategies in networks [8][9][10]. Many approximate methods applied to the SIS model have proposed various types of estimates for τ c .…”

Complex Networks Analysis by Spectral Graph Theory

“…It can be concluded that the lower spectral radius corresponds to greater robustness in the network in terms of spreading viruses and greater protection from viruses can be achieved through minimization of the spectral radius [5].…”