The viral conductance of a network

Besides the epidemic threshold, the recently proposed viral conductance ψψ by Kooij et al. [11] may be regarded as an additional characterizer of the viral robustness of a network, that measures the overall ease in which viruses can spread in a particular network. Motivated to explain observed features of the viral conductance ψψ in simulations [29], we have analysed this metric in depth using the NN-intertwined SIS epidemic model, that upper bounds the real infection probability in any network and, hence, provides safe-side bounds on which network protection can be based. Our study here derives a few exact results for ψψ, a number of different lower and upper bounds for ψψ with variable accuracy. We also extend the theory of the NN-intertwined SIS epidemic model, by deducing formal series expansions of the steady-state fraction of infected nodes for any graph and any effective infection rate, that result in a series for the viral conductance ψψ. Though approximate, we illustrate here that the NN-intertwined SIS epidemic model is so far the only SIS model on networks that is analytically tractable, and valuable to provide first order estimates of the epidemic impact in networks. Finally, inspired by the analogy between virus spread and synchronization of coupled oscillators in a network, we propose the synchronizability as the analogue of the viral conductance.