Statistical Model for Soical Network

Transcription

1 Statistical Model for Soical Network Tom A.B. Snijders University of Washington May 29, 2014

2 Outline 1 Cross-sectional network 2 Dynamic s

3 Outline Cross-sectional network 1 Cross-sectional network 2 Dynamic s

4 Outline Cross-sectional network 1 Cross-sectional network 2 Dynamic s

5 Conditionally Uniform Models Holland and Leinhardt (1976), Wasserman (1977) E.g, U E, U M, A, N distribution The conditioning statistics contain that which is relevant in the studied phenomena, and the rest is randomness. Conditionally uniform distributions are typically used as straw man null hypotheses Network properties that the researchers wishes to control for are put in the conditioning statistic, and the theory that is put to the test is expressed by a different statistic, for which then the p-value is calculated under the conditionally uniform distribution.

6 Limitations The conditional distribution is hard to obtain, e.g, in-degree & out-degree Rejection of the null hypothesis does not provide a first step toward constructing a model for the phenomenon being studied

7 Outline Cross-sectional network 1 Cross-sectional network 2 Dynamic s

8 Cross-sectional network : Assumes the existence of latent (i.e., unobserved) variables, such that the observed variables have a simple probability distribution given the latent variables.

9 Discrete space Cross-sectional network Discrete space: Stochastic block model: Holland et al. (1983), Snijders and Nowicki (1994),Nowicki and Snijders (2001), and Daudin et al. (2008), Mixed Membership:Airoldi et al.(2008) Pr(Y ij = a X = x) = η a (x i, x j )

10 Distance model Distance model: Hoff et al.(2002), Handcock et al. (2007) Pr(Y ij = 1) = π(d(i, j)) Ultrametric Space, Freeman (1992) Euclidean Space, Hoff(2002) Mixture model, Handcock et al.(2007)

11 Sender and receiver Sender and receiver: Holland and Leinhardt (1981),van Duijn et al.(2004), Hoff (2005) p 1 (y) = P (Y = y) = exp{ρm + θx ++ + i α ix i+ + j β jx +j } Each actor has two parameters α i,β i responsible, respectively, for the tendency of the actor to send ties (âactivityâ, influencing the out-degrees) and the tendency to receive ties (âpopularityâ, influencing the in-degrees) In addition there are parameters influencing the total number of ties and the tendency toward reciprocation The large number of parameters, two for each actor, is a disadvantage of this model Hoff(2005) : random sender effects A i, receiver effects B j and reciprocity effects C ij = C ji, bilinear effects D i D j

12 Ordered Space Cross-sectional network Ordered Space De Vries (1998) Mogapi(2009) probabilities of ties depend on how the points are ordered, together with covariates. π 1 + β z ij if i j logit(p (Y ij = 1) = π 2 + β z ij if j i π 3 + β z ij if i j and j i

13 Outline Cross-sectional network 1 Cross-sectional network 2 Dynamic s

14 Markov conditional independence: Frank, Strauss (1986), Wasserman, Pattison (1996) Paritial conditional independence: Pattison, Robins (2002), Snidjer et.al (2006), Hunter (2007) Exponential Random Network Models: Fellows(2012)

15 Overview: Conditional Independence Assumptions Conditionally uniform models condition on observed statistics, and try to assess whether these are sufficient to represent the observed network Latent space models postulate the existence of a space in which the nodes occupy unobserved (latent) positions, such that the tie indicators are independent conditionally on these positions. s, when using subgraph counts as sufficient statistics, are based on conditional independence assumptions between the observed tie variables.

16 Outline Cross-sectional network Dynamic s 1 Cross-sectional network 2 Dynamic s

17 Outline Cross-sectional network Dynamic s 1 Cross-sectional network 2 Dynamic s

18 Dynamic s Snijders and van Duijn (1997), Snijders (2001) Steglich et.al (2010): with behavior co-evolution

19 Actor-oriented model Cross-sectional network Dynamic s Intention: model the evolution of the network as driven by the actors efforts to seek better network configurations Goal: find what and how the factors drive the actors during network evolution. Estimate the unknown coefficients θ which maximize momentary total objective function f i (θ, x) + g i (θ, x, j) + ɛ i (t, x, j) objective function gratification function random error Assumes the following behavior of the actors: Control : Every actor has complete control over his or her out-going ties. individual change rate Relative myopia : The decisions of the actors are based only on the present state and the states that can be reached by a single change to their composition. Markov property Complete information : Each actor is assumed to have full knowledge about the state of the network at each given time. Re-evaluate the network after every change

20 Dependence Structure Dynamic s For actor i, and j Dependence Structure Coefficient activity: s i1(x) = j x ij β 1 reciprocity: s i2(x) = j x ijx ji β 2 citation attraction: s i3(x) = j x ijw j β 3 citation dissimilarity: s i4(x) = j x ij w i w j β 4 transitivity: s i5(x) = j,h x ijx ih x jh β 5

21 Dynamic s Continuous Time Markov Chain Model K objective function f i(θ, x) = β k s ik (x) k=1 r(θ, i, j, x) = f i(θ, x(i j)) + g i(θ, x, j), g i 0 ministep x y = e N x e y e = 1 rate function q(θ, x, y) = λ(θ, x)p(θ, x, y) individual change rate λ(θ, x) = i V λ i(θ, x) λ i(x, θ) = ρ{e α + si1(x) n 1 (e α e α )} actor i went to a coffee shop λ i(x, θ)/λ(x, θ) bought or spilled a cup of coffee on j e r(θ,i,j,x) k V \{i} er(θ,i,k,x) θ = (ρ, α, β 1,..., β k )

22 Outline Cross-sectional network Dynamic s 1 Cross-sectional network 2 Dynamic s

23 Dynamic s Dynamic s Robins and Pattison (2001), Hanneke et al. (2010): Discrete Temporal Krivitsky(2014): different social selection and social influence mechanisms Almquist and Butts(2013): lagged network logistic regression

24 Outline Cross-sectional network Dynamic s 1 Cross-sectional network 2 Dynamic s

25 Dynamic s Guo F et al. (2007) : Hidden Temporal ERGM Xing et al. (2010) : State-space Mixed Membership Block Dynamic Model Sarkar and Moore (2005) : Longitudinal Westveld and Hoff (2011) : Mixed Effect Model