The viral conductance of a network

Besides the epidemic threshold, the recently proposed viral conductance w 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 w in simulations [29], we have analysed this metric in depth using the N-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 w, a number of different lower and upper bounds for w with variable accuracy. We also extend the theory of the N-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 w. Though approximate, we illustrate here that the N-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. We investigate the influence of the network topology on the spread of viruses, digital as well as biological ones, whose dynamics is modeled by a susceptible-infected-susceptible (SIS) type of process [2]. Digital viruses (of all kind, and generally called mal-ware) are living in cyberspace and use mainly the Internet as the transport media, while biological viruses contaminate other living beings on earth and use ''contacts'' among their victims as their propagation networks. The increasing threats from cybercrime and the expected outbreak of new lethal, biological viruses justify studies on virus spread in graphs. In particular, network operators are interested to know (a) how vulnerable their network is for epidemics and (b) how to protect or modify their infrastructure, most often, at minimal cost (see e.g. [10]). In contrast to the negative connotation of a threat, information spread (news, rumors, etc.) in the new types of on-line social networks like Facebook, Twitter, Digg [20], etc. resembles epidemic diffusion. While this analogy still needs to be verified and evaluated via extensive measurements , epidemic theory is expected to be fruitful in assessing the …