Given a network of who-contacts-whom or who-links-to-whom, will a contagious virus/product/meme spread and 'take-over' (cause an epidemic) or die-out quickly? What will change if nodes have partial, temporary or permanent immunity? The epidemic threshold is the minimum level of virulence to prevent a viral contagion from dying out quickly and determining it is a fundamental question in epidemiology and related areas. Most earlier work focuses either on special types of graphs or on specific epidemiological/cascade models. We are the first to show the G2-threshold (twice generalized) theorem, which nicely de-couples the effect of the topology and the virus model. Our result unifies and includes as special case older results and shows that the threshold depends on the first eigenvalue of the connectivity matrix, (a) for any graph and (b) for all propagation models in standard literature (more than 25, including H.I.V.). Our discovery has broad implications for the vulnerability of real, complex networks, and numerous applications, including viral marketing, blog dynamics, influence propagation, easy answers to 'what-if' questions, and simplified design and evaluation of immunization

Given a contact network that changes over time (say, day vs night connectivity), and the SIS (susceptible/infected/susceptible, flu like) virus propagation model, what can we say about its epidemic threshold? That is, can we determine when a small infection will "take-off" and create an epidemic? Consequently then, which nodes should we immunize to prevent an epidemic? This is a very real problem, since, e.g. people have different connections during the day at work, and during the night at home. Static graphs have been studied for a long time, with numerous analytical results. Time-evolving networks are so hard to analyze, that most existing works are simulation studies [5]. Specifically, our contributions in this paper are: (a) we formulate the problem by approximating it by a Non-linear Dynamical system (NLDS), (b) we derive the first closed formula for the epidemic threshold of timevarying graphs under the SIS model, and finally (c) we show the usefulness of our threshold by presenting efficient heuristics and evaluate the effectiveness of our methods on synthetic and real data like the MIT reality mining graphs.

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Abstract-In this paper, we study the intertwined propagation of two competing "memes" (or data, rumors, etc.) in a composite network. Within the constraints of this scenario, we ask two key questions: (a) which meme will prevail? and (b) can one influence the outcome of the propagations? Our model is underpinned by two key concepts, a structural graph model (composite network) and a viral propagation model (SI1I2S). Using this framework, we formulate a non-linear dynamic system and perform an eigenvalue analysis to identify the tipping point of the epidemic behavior. Based on insights gained from this analysis, we demonstrate an effective and accurate prediction method to determine viral dominance, which we call the EigenPredictor. Next, using a combination of synthetic and real composite networks, we evaluate the effectiveness of various viral suppression techniques by either a) concurrently suppressing both memes or b) unilaterally suppressing a single meme while leaving the other relatively unaffected.

Abstract. Short-range, point-to-point communications for mobile users enjoy increasing popularity, particularly with the rise in Bluetooth -equipped mobile devices. Unfortunately, virus writers have begun exploiting lax security in many mobile devices and subsequently developed malware exploiting proximity-based propagation mechanisms (e.g. Cabir or CommWarrior). So, if given an ad-hoc network of such mobile users, will a proximity-spreading virus survive or die out; that is, can we determine the "tipping point" between survival and die out? What effect does the average user velocity have on such spread? We answer the initial questions and more. Our contributions in this paper are: (a) we present a framework for analyzing epidemic spreading processes on mobile ad hoc networks, (b) using our framework, we are the first to derive the epidemic threshold for any mobility model under the SIS model, and (c) we show that the node velocity in mobility models does not affect the epidemic threshold. Additionally, we introduce a periodic mobility model and provide evaluation via our framework. We validate our theoretical predictions using a combination of simulated and synthetic mobility data, showing ultimately, our predictions accurately estimate the epidemic threshold of such systems.