Assessing network robustness under SIS epidemics: The relationship between epidemic threshold and viral conductance

Telecommunication networks, as well as other network types, are critical infrastructures where any service disruption has a notable impact on individuals. Hence, studying network dynamics under failures or attacks is of paramount importance. In this paper, we assess the robustness of networks with respect to the spread of Susceptible-Infected-Susceptible (SIS) epidemics, using the N-Intertwined Mean-Field Approximation (NIMFA). A classical robustness metric is the NIMFA epidemic threshold, which is inversely proportional to the largest eigenvalue of the adjacency matrix, also called the spectral radius. Besides the NIMFA epidemic threshold, the viral conductance has been proposed as a measure incorporating the average fraction of infected nodes in the steady state for all possible effective infection rates. In general, the viral conductance provides more information about the network's behavior with respect to virus spreading, however, the full picture is not always necessary. The aim of this paper is to understand when the spectral radius is adequate for reflecting robustness. By analyzing the relationship between spectral radius and viral conductance in several graph classes, we show that the two metrics are highly correlated. We thus conclude that the spectral radius is sufficient to compare the robustness of networks belonging to the same class.