Threshold Conditions for Arbitrary Cascade Models on Arbitrary Networks

Abstract

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

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“…Our results for all the mobility models discussed in this paper hold for all the epidemic models covered in [24] as well.…”

“…Given recent results on epidemic thresholds on static networks [24], we believe that our results will carry through for many other epidemic models as well e.g. SIR (mumps-like), SIRS, SEIR, MSEIR etc [13] which capture differences between the way various diseases spread.…”

Average is Boring: How Similarity Kills a Meme's Success

“…the characteristics of the product, plays a major role in the final outcome of the product success. Thus, our results can be used as complementary approach to the one studied by Weng et al Many other papers focused on the role of social networks in meme popularity, whether to model cascade of information1018 or to identify the users in a position of power1945. These models share with Weng et al the focus on the social network, which we proved not to be the sole source of information to explain a meme's popularity.…”

“…The evolution of our communication power has not only increased our cultural production capability, but also its tracking, studying and understanding of that power6141617181920. The challenge here is to quantitatively and objectively measure the potential reach and impact of ideas and behaviours.…”

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“…In this context, we have seen theoretical developments of these models [8], [9], focusing particularly in their asymptotic behavior. A number of applications of such models were also explored [10], [11], including works investigating relevant properties of epidemic models on real networks [12].…”