Diffusion in Networks Dr. Henry Hexmoor Department of Computer Science Southern Illinois University Carbondale 1/17/2019

Diffusion The way in which an individual’s choices depend on what other people do. Modeling the processes by which new ideas and innovations are adopted by a population. 1/17/2019

𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 ×𝐹 𝑡−1 The Bass Model F(t) = Fraction of population who have adopted a new idea or product. p = rate of innovation (lead, generate new ideas) q = rate of imitation (follow, imitate) 𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 ×𝐹 𝑡−1 Innovation Imitation 1/17/2019

𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 1−𝐹 𝑡−1 ×𝐹 𝑡−1 The Bass Model Innovation 𝐹 𝑡 =𝐹 𝑡−1 +𝑝 1−𝐹 𝑡−1 +𝑞 1−𝐹 𝑡−1 1−𝐹 𝑡−1 ×𝐹 𝑡−1 𝑑𝐹(𝑡) 𝑑𝑡 = 𝑝+𝑞 ×𝐹 𝑡 1−𝐹 𝑡 𝐼𝑓 𝐹 0 = 0 𝑎𝑛𝑑 𝑝 > 0 𝑡ℎ𝑒𝑛, 𝐹 𝑡 = 1 − 𝑒 − 𝑝+𝑞 𝑡 1+ 𝑞 𝑝 𝑒 − 𝑝+𝑞 𝑡 Imitation Mostly adopted, e.g., Soda Rarely adopted 1/17/2019

S: Susceptible, I: Infected , R: Removed/Recovered. Epidemic Models They are similar to each other and all use the infection life cycle, e.g, SIR, SIS, SI and SIRE. S: Susceptible, I: Infected , R: Removed/Recovered. SIR SIS 1/17/2019

SIR Model A disease that is caught once in a node’s lifetime. Divide population into S and I. Each I node will remain I for 𝑡 1 time period. During 𝑡 1 time steps, each node has probability, p, of passing disease to on S neighbor. After 𝑡 1 expires, all I nodes become R. 1/17/2019

SIR Model Contagiousness is constant at each step of I phase. Contagion probability is uniformly fixed to p. Certain links may provide a more rapid contagion due to nature of contacts. SIR can be interpreted as a percolation model if and only if links are pipes. SIR saturates after time - no more infections. 1/17/2019

SIS Model A disease that is caught repeatedly by a node. Divide population into S and I. Each I node will remain I for 𝑡 1 time period. During 𝑡 1 time steps, each node has probability, p, of passing disease to on S neighbor. After 𝑡 1 expires, each I node becomes S, as before. In SIS, if all nodes are simultaneously free of disease (S), then the epidemic has ended. 1/17/2019

The Cascade Model Can be captured by using a coordination game. Consider v and w are neighbors playing the given coordination game. Let p be the fraction of v’s neighbors playing A and (1−𝑝) fraction play B. Let v have d neighbors. v’s payoff for 𝐴=𝑝 ×𝑑×𝑎 𝐵= 1−𝑝 ×𝑑×𝑏 Behavior A is preferred Iff 𝑝𝑑𝑎≥ 1−𝑝 𝑑𝑏 which is equivalent to 𝑝≥ 𝑏 𝑎+𝑏 =𝑞 v’s preference threshold 1/17/2019

The Cascade Model 𝑝≥ 𝑏 𝑎+𝑏 =𝑞 If at least q fraction of its neighbors adopt A, v adopts A as well. In other words, if q is small, v adopts A. If q is large, then v adopts B. Behavior adoption spreads in the network by successive behavior adoption. This is the basic for viral spreading. 1/17/2019

A Cluster of Density p A set of nodes such that each node in the set has at least a fraction p of its neighbors in the set is a cluster of density p. A cluster of density 1 is all the nodes in the network. For each i and j clusters of density p, i ∪ j is also a cluster of density p. A cascade halts when it encounters a dense cluster. Clusters are the only obstacles to cascades. 1/17/2019

Cascade Model Revised Individuals (i.e., v and w) value behaviors A and B differently. If v has d neighbors of when p fraction have behavior A, (1−𝑝) have behavior B. If 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 ℎ𝑎𝑣𝑒 𝐴 𝑑 =𝑝≥ 𝑏 𝑣 𝑎 𝑣 + 𝑏 𝑣 = 𝑞 𝑣 (v’s threshold for adopting A), then A is the better choice. 1/17/2019

An Example 1 3 since 0.1 < 1/3. Then, 3 2, 3 5, 2 4, 5 6 Cascade stops at node 4 and 6. 1/17/2019

References D. Easly, J. Kleinberg, 2010. Networks, Crowds, and Markets, Cambridge University press. 2. M. Jackson, 2008. Social and Economic Networks, Princeton University press. 1/17/2019