Consider our model of the diffusion of a new behavior through a social network. Recall that...

 

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Consider our model of the diffusion of a new behavior through a social network. Recall that to define this model, we have a network, a behavior B that everyone starts with, and a threshold q for switching to a new behavior A. Any node will switch to A if at least a fraction q of its neighbors have adopted A. If exactly q of its neighbors have adopted A, we assume it adopts A, too. Consider the network depicted below; each node starts with the behavior B. F D A B H Q3. Now suppose that the two shaded nodes (D and I) both adopt the new behavior A, and each other node has a threshold of q = 1/2 for switching to behavior A. a) If other nodes follow the threshold rule for choosing behaviors, which nodes will eventually switch to the new behavior A? You do not have to explain your answer. (2 points) b) Describe and explain a possible history of the system (which nodes adopt at the first time step, which nodes adopt at the second time step, etc.). (3 points) c) Name a cluster of density greater than 1-q=1/2 that blocks behavior A from spreading to all nodes, starting from D and I, at threshold q. (2 points) d) Explain why the density of this cluster is greater than 1-q = 1/2. (3 points) Consider our model of the diffusion of a new behavior through a social network. Recall that to define this model, we have a network, a behavior B that everyone starts with, and a threshold q for switching to a new behavior A. Any node will switch to A if at least a fraction q of its neighbors have adopted A. If exactly q of its neighbors have adopted A, we assume it adopts A, too. Consider the network depicted below; each node starts with the behavior B. F D A B H Q3. Now suppose that the two shaded nodes (D and I) both adopt the new behavior A, and each other node has a threshold of q = 1/2 for switching to behavior A. a) If other nodes follow the threshold rule for choosing behaviors, which nodes will eventually switch to the new behavior A? You do not have to explain your answer. (2 points) b) Describe and explain a possible history of the system (which nodes adopt at the first time step, which nodes adopt at the second time step, etc.). (3 points) c) Name a cluster of density greater than 1-q=1/2 that blocks behavior A from spreading to all nodes, starting from D and I, at threshold q. (2 points) d) Explain why the density of this cluster is greater than 1-q = 1/2. (3 points)