Diffusion of Innovation

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Diffusion of Innovation Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Social Network Analysis MAGoLEGO course Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 1 / 22

Lecture outline 1 Diffusion of innovation 2 Influence propagation models Linear threshold model 3 Influence maximization problem Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 2 / 22

Diffusion process Propagation process: Information based models: - ideas, knowledge - virus and infection - rumors, news Decision based models: - adoption of innovation - joining politcal protest - purchase decision Local individual decision rules will lead to very different global results. microscopic changes macroscopic results Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 3 / 22

Ryan-Gross study Ryan-Gross study of hybrid seed corn delayed adoption (after first exposure) Information effect vs adopting of innovation Ryan and Gross, 1943 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 4 / 22

Ryan-Gross study Hybrid corn adoption Percentage of total acreage plated Griliches, 1957 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 5 / 22

Diffusion of innovation Everett Rogers, Diffusion of innovation book, 1962 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 6 / 22

Bass Diffusion model Frank Bass, 1969, A new product growth model for consumer durables Growth model of how new products get adopted Two types of agents, two key parameters: - p - innovation or spontanious adoption rate (coefficient of innovation) - q - rate of immitation (coefficient of immitation) Let F (t) fraction of agents adopted by time t F (t + 1) = F (t) + p(1 F (t))δt + q(1 F (t))f (t)δt df (t) dt = (p + qf (t))(1 F (t)) Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 7 / 22

Bass Diffusion model Soltion of Bass model - S-curve. When F (0) = 0 F (t) = 1 e (p+q)t 1 + q p e (p+q)t Emprical p 0.01 0.03, q 0.3 0.5, when t in years Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 8 / 22

Diffusion of innovation What influences potential adopters: relative advantage of the innovation compatibility with current ways of doing things complexity of the innovation triability - the ease of testing observability of results Some questions remain: how a new technology can take over? who different technologies coexist? what stops new technology propagation? Everett Rogers,1962 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 9 / 22

Network structure From the population level to local structure Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 10 / 22

Network coordination game Local interaction game: Let u and v are players, and A and b are possible strategies Payoffs if u and v both adopt behavior A, each get payoff a > 0 if u and v both adopt behavior B, each get payoff b > 0 if u and v adopt opposite behavior, each get payoff 0 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 11 / 22

Threshold model Network coordination game, direct-benefit effect Node v to make decision A or B, p - portion of type A neighbors to accept A: a p d > b (1 p) d Threshold: p b/(a + b) q = Accept new behavior A when p q b a + b Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 12 / 22

Cascades Cascade - sequence of changes of behavior, chain reaction Let a = 3, b = 2, threshold q = 2/(2 + 3) = 2/5 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 13 / 22

Cascade propagation Let a = 3, b = 2, threshold q = 2/(2 + 3) = 2/5 Start from nodes 7,8: 1/3 < 2/5 < 1/2 < 2/3 Cascade size - number of nodes that changed the behavior Complete cascade when every node changes the behavior Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 14 / 22

Cascades and clusters Group of nodes form a cluster of density ρ if every node in the set has at least fraction ρ of its neighbors in the set Both clusters of density ρ = 2/3. For cascade to get into cluster q 1 ρ. images from Easley & Kleinberg Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 15 / 22

Linear threshold model Influence comes only from NN N(i) nodes, w ij influence i j Require j N(i) w ji 1 Each node has a random acceptance threshold from θ i [0, 1] Activation: fraction of active nodes exceeds threshold w ji > θ i active j N(i) Initial set of active nodes A o, iterative process with discrete time steps Progressive process, only nonactive active Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 16 / 22

Cascades in random networks multiple seed nodes (a) Empirical network; (b), (c) - randomized network P. Singh, 2013 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 17 / 22

Influence maximization problem Initial set of active nodes A o Cascade size σ(a o ) - expected number of active nodes when propagation stops Find k-set of nodes A o that produces maximal cascade σ(a o ) k-set of maximum influence nodes NP-hard D. Kempe, J. Kleinberg, E. Tardos, 2003, 2005 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 18 / 22

Influence maximization Greedy maximization algorithm: Given: Graph and set size k Output: Maximum influence set A 1 Select a node v 1 that maximizes the influence σ(v 1 ) 2 Fix v 1 and find v 2 such that maximizes σ(v 1, v 2 ) 3 Repeat k times 4 Output maximum influence set: A = {v 1, v 2...v k } Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 19 / 22

Experimental results Linear threshold model network: collaboration graph 10,000 nodes, 53,000 edges Greedy algorithm finds a set S such that its influence set σ(s) is σ(s) (1 1 e )σ(s ) from the true optimal (maximal) set σ(s ) D. Kempe, J. Kleinberg, E. Tardos, 2003 Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 20 / 22

References Contagion, S. Morris, Review of Economic Studies, 67, p 57-78, 2000 Maximizing the Spread of Influence through a Social Network, D. Kempe, J. Kleinberg, E. Tardos, 2003 Influential Nodes in a Diffusion Model for Social Networks, D. Kempe, J. Kleinberg, E. Tardos, 2005 A Simple Model of Global Cascades on Random Networks. D. Watts, 2002. Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 21 / 22

Course summary 1 Introduction to network science 2 Descriptive network analysis 3 Mathematical models of networks 4 Node centrality and ranking on networks 5 Network communitiee 6 Network structure and visualization 7 Epidemics and information spread 8 Diffision of innovation Leonid E. Zhukov (HSE) Lecture 8 05.06.2015 22 / 22

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