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Consider a discrete time model and let f t be the

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Unformatted text preview: ffects. Consider a discrete-time model and let F (t ) be the fraction of agents infected at time t . The Bass model is described by the difference equation: F (t ) = F (t − 1) + p (1 − F (t − 1)) + q (1 − F (t − 1))F (t − 1). The term p (1 − F (t − 1)) is the infection rate times the fraction of uninfected agents. The term q (1 − F (t − 1))F (t − 1) is the contagion rate times the frequency of encounters between healthy and infected agents. 4 Networks: Lecture 8 Bass Model–2 A continuous time version of this model is described by d F (t ) = (p + qF (t ))(1 − F (t )), dt with F (0) = 0. This is a nonlinear differential equation, but admits a closed form solution F (t ) = 1 − e − (p +q )t . 1 + q e − (p +q )t p Note that the levels of p and q scale time, the ratio of q to p determines the overall shape of the curve. 5 Networks: Lecture 8 Bass model–3 Figure: Diffusion curves: left is for p < q and right is for p > q . Many empirical studies have found diffusion patterns that are S-shaped (e.g. adoption of hybrid corn seeds among Iowa farmers). Let us interpret this in the context of adoption of new technologies. First adopters are almost entirely those who adopt from their spontaneous ˙ innovation (when F (t ) is close to 0, F (t ) = p ). As process progresses, there are more agents to be imitated leading to an increase in the rate of diffusion, which eventually slows down since there are fewer agents to do the imitating. 6 Networks: Lecture 8 Diffusion in a Network with Immune Nodes The problems of modeling contagion or the spread of information through a society involve determining “when paths exist that connect different nodes”, i.e., understanding the component structure. Let us consider the following problem. There is a society of n individuals. Initially one of them is infected with a disease. Each individual is immune with probability π . The question of whether the disease can spread to a nontrivial fraction of the population amounts to whether the infected...
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