Presented by:

1. Contrasting multiple social network autocorrelations for binary outcomes ,with application to technology adoption BIN ZHANG, Temple UniversityANDREW C. THOMAS, Carnegie Mellon UniversityPATRICK DOREIAN, University of Pittsburgh and University of LjubljanaDAVID KRACKHARDT and RAMAYYA KRISHNAN Presented by: Smit ambalia

2. Contents…. • Introduction • Background • Method • Model specification • Expectation-maximization solution • Full Bayesian solution • Sensitivity to prior specification • Application • Auto purchase data of Yang and Allenby[2003] • Caller ring back tone usage in a mobile network • Conclusion

3. Introduction • Social network is one such complexity • Researchers once assumed that an individual’s decision to purchase a product or adopt a technology , is associated with their characteristics such as age,education,income [Allenby and Rossi 1998;kamakura and Russell 1989] • Investigations into behavior in network is limited to people, Analyzing the behavior of individuals , such as their purchasing or technology adoption tendencies requires statistical data • A model with explicit autocorrelation between individual outcomes , defined with a single network structure term • Data shows that members are observed to be of multiple networks like college network, family network, all networks have some connection to the outcome of interest , so model condenses all networks into one relation will be sufficient • Models have been developed to include two or more network autocorrelation terms such as binary outcomes [Doreian 1989]

4. Introduction • Uses two approaches for hierarchical model • Expectation-maximization Algorithm • Markov Chain Monte Carlo(MCMC), method from Bayesian statistics • Construct algorithm in R programming language and validate the MCMC method with the posterior quintiles method f

5. Background • Network models have been used to study diffusion since the Bass [1969] model, a population approach assumes that everyone has in social network has same probability of interacting • y = Xβ + θ, θ = ρWθ + e, • Standard network autocorrelationmodelsuch as thoseBurt[1987] and Leenders [1997], can only accomodate one network , but actor isalwaysunderinflunced of multiple network • So, none of thesemodels are adequate , a modelthat can acomodate more then one networksisnecessary • Cohension and structuralquivalence are two social networksthatexplaindiffusion of innovation • y = Xβ + ρ1W1y + ρ2W2y + e,

6. Background(Continue…. ) • Model takes a continues dependent variable ,Fujimoto and valente [2011] present plausible solution for binary outcomes by directly inserting an autocorrelation term Wyinto right side of logistic regression • Although it may support a binary dependent variable and multiple network term, this model doesn’t satisfy assumption of logistic regression- the observations are not conditionally independent • Yand and Allenby [2003] propose hierarchical Bayesian auto regression mixture model to analyze the effect of multiple network autocorrelation

7. Method • Propose a new auto-probit model , which accommodates multiple regimes of network autocorrelation terms for the same group of actors , which we call the multiple auto-probit(m-NAP) • We provide two methods to obtain estimates for our model • Expextation-Maximization : employs maximum likelihood approach • Markov chain montecarlo : treats model as Bayesian

8. Model Specification • Actors are assumed to have k different types of network connection between them, where Wi, ith network , y is length n of observed binary choices , and is an indicator function of the latent preferences consumers z

9. Model Specification (Continue..) • Our model explicitly allows multiple competing networks that can be by different network mechanisms on a exciting basis of network ties • For Example , W1 describes effect acting directly on a declared tie , such as social influence • Whereas W2 describes structural equivalence due to those ties

10. Expectation-maximization Solution • Develop an approach by maximizing the likelihood of the model obtain the maximum likelihood estimate for the model using EM • Method consists of two steps : first, we estimate the expected value of functions of unobserved z given the current parameters set Φ, Second , We use these estimates to form a complete data set {y,X,z}, with which we estimate s new Φ by maximizing the expectation of likelihood of complete data

11. Expectation-maximization Solution • e-step • In M-step , we maximize G(φ/φ(t) to get B (t+1),p(t+1), σ(t+1), For the next step • We replace  φ(t) to  φ(t+1) and repeat the e-step and m-step until all the parameters converge. Parameter estimates from the em algorithm converge to the MLE estimates [Wu 1983]

12. Expectation-maximization Solution

13. Full Bayesian Solution • Since the observed choice of a consumer is decided by hi/her unobserved preferences , this model is hierarchical structure , so it’s hierarchical Bayesian method • With mirkov Chain Monte Carlo , we generate draws from series of full condition probability distributions , derived from joint distributions

14. Validation of Bayesian Software • This method is error free implementation , have high complexity , lack of software causes many researchers to develop their own, greatly increase the chance of software error • Many models are not validated , many of them have errors and don’t correct estimations , and confirms that code returns correct result

16. Application • Auto Purchase Data of Yang and Allenby [2003] : • Yang’s [2003] Japanese car data to compare the findings of our method with those in the original study • Important Question of interest is whether the preferences for Japanese car among consumers are independent or not , is measured by geographical location , Wij=1 if consumer I and share same Zip Code otherwise 0 • Explanatory variables include actors’ demographic information such as age, ethic group , annual household income • To construct network, Yang and Allenby use the fact that the consumers’ home addresses are in same Zip code as indicator of a connection

17. Application • By comparing the parameter of Yang and Allenby’s model to these for n-NAP on the same dataset, with the same underlying definition of network structure , demonstrate the value of separating the impact of various network autocorrelation • Specify second network term W2 to be structural equivalence of two consumers, calculated as simple adjutancy distance between two vectors representing individuals’ connection to other individuals in the network

19. Caller Ring-Back tone usage in a mobile network • Use m-Nap to investigate the purchase of caller Ring back Tones (CRBT) within a cellular phone network , • When someone call, the caller hear the song ,jock or other massages chosen by subscribers until he answer the phone • From Original network, we derive two matrices corresponding to cohesion and structural equivalence • Cohesion assumes caller who make calls to each other will hear the caller party’s CRBT , if may get interested in CRBT and eventually adopt the technology

20. Caller Ring-Back tone usage in a mobile network • Normalize the cohesion matrix by dividing each row by total no of adopters , to make the matrix element the percentage of adoption • Structural equivalence is defined as the adjacency distance between two caller • There is less clear that there is an obvious mechanism for how structural equivalence can impact adoption , as it relates to a relationship that does not expose the caller to the CRBT

21. Conclusion • Introduced new auto-probit model to study binary choice of a group of actors that have multiple network relationships among them. • Model compare influence from different networks on binary choices of individuals , For example weather to adopt a new technology or purchase a new product or not when they are embedded in multiple types of networks • We found that EM solution cannot estimate the parameters for this particular model , only hierarchal Bayesian solution can be used here • Finally , we compare the estimates returned by Yang and Allenby , NAP with one network effect (Cohesion) , and NAP with two network effects (Cohesion and structural equivalence ) by using real data

22. References • Agarwal, R., Gupta, A. K., And kraut, R. 2008. Editorial overview—the interplay between digital and social networks. Inform. Syst. Res. 19, 3, 243–252 • Allenby, G. M. And rossi, P. E. 1998. Marketing models of consumer heterogeneity. J. Econometrics 89, 1–2, 58-78 • Anselin, L. 1988. Spatial econometrics: methods and models. Springer • Bass, F. M. 1969. A new product growth for model consumer durables. Manage. Sci. 15, 5, 215–227.

23. Any Questions ? Thank you