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We have studied this problem when the underlying

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Unformatted text preview: individual lies in a “giant component” of the network with the immune nodes removed. We have studied this problem when the underlying network is an Erd¨s-Renyi network. o We will now generalize to arbitrary degree distributions. 7 Networks: Lecture 8 Diffusion with General Degree Distributions Recall that the degree distribution of a neighboring node is given by dP (d ) ˜ P (d ) = . �d � From this, we showed that the expected number of children is given by �d 2 � − �d � ˜ E[number of children] = . �d � Hence the expected number of infected children (basic reproductive number of the disease) is �d 2 � − �d � λ ≡ (1 − π ) . �d � 8 Networks: Lecture 8 Diffusion with General Degree Distributions Recall the branching process analysis: If λ < 1, then with probability one, the disease dies out after a finite number of stages. If λ > 1, then with positive probability, the disease persists by infecting a large portion of the population. This yields the following threshold for the probability of immune π : a giant �d 2 � − 2�d � component will emerge if π< , �d 2 � − �d � i.e., if the fraction of immune nodes is below this threshold. ¯ For example, for a regular network (each node with degree d ), this leads to ¯ d −2 π= ¯ . d −1 ¯ If d = 2, giant component never emerges. ¯ If d = 3, giant component emerges if < half the population is immune. For the Erd¨s-Renyi graph, we have �d 2 � = �d �2 + �d � and �d � = (n − 1)p . o 1 This yields the threshold π = 1 − (n−1)p or p (n − 1)(1 − π ) = 1, as before. 9 Networks: Lecture 8 Diffusion with General Degree Distributions For a power-law degree distribution (or a scale-free network) with P (d ) ∼ d −γ , γ < 3, we have that the �d 2 � is diverging in n. Therefore, the contagion threshold for this case is π = 1, i.e., all nodes have to be immune before the giant component of susceptible nodes di...
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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.

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