Generalized Exponential Random Graph Models: Inference for Weighted Graphs

Transcription

1 Generalized Exponential Random Graph Models: Inference for Weighted Graphs James D. Wilson University of North Carolina at Chapel Hill June 18th, 2015 Political Networks, 2015 James D. Wilson GERGMs for Weighted Networks 1 / 30

2 Outline 1 Setting the Stage: ERGMs for Unweighted Graphs 2 Generalized Exponential Random Graphs (GERGMs) Model Inference via MCMC Applications and Degeneracy Case Study 3 Tutorial on gergm.r Software 4 Using gergm.r in person James D. Wilson GERGMs for Weighted Networks 2 / 30

3 Networks / Graphs Means to visualize, model, and analyze interactions of a complex system Treat an actor" of the system as a vertex and place edges between actors that interact James D. Wilson GERGMs for Weighted Networks 3 / 30

4 Exponential Random Graph Models (ERGMs) Setting: Network x composed of n nodes and M edges Assume binary edges between nodes i, j: x i,j {0, 1} X: family of all graphs with n nodes and binary edges P X : probability measure on X Aim: Identify (and then estimate) the relational covariates that capture network structure through P X James D. Wilson GERGMs for Weighted Networks 4 / 30

5 Exponential Random Graph Models (ERGMs) The probability (likelihood) of observing network x: P X (x, θ) = exp(θ h(x)) exp(θ, x {0, 1}M h(z)) z X h {0, 1} M R p : Network covariates θ R p : unknown parameters (we have to estimate these!) James D. Wilson GERGMs for Weighted Networks 5 / 30

6 ERGM: Covariate Specification Endogeneous: network-level Exogeneous: relational data-driven covariates James D. Wilson GERGMs for Weighted Networks 6 / 30

7 ERGMs: Food for Thought Model selection Goodness of fit diagnostics Computational tools for estimation Markov Chain Monte-Carlo estimation Maximum pseudo-likelihood estimation Likelihood degeneracy: how can we avoid trivial models? James D. Wilson GERGMs for Weighted Networks 7 / 30

8 Weighted Graphs Setting: Network y composed of n nodes and M edges Edges are continuous valued between nodes i, j: y i,j (, ) Y : family of all graphs with n nodes P Y : probability measure on Y Examples: Correlation networks Financial Lending Networks Voting Networks Brain Networks James D. Wilson GERGMs for Weighted Networks 8 / 30

9 Inference for weighted networks GERGM: Generative exponential model for y that incorporates network covariates population edge-level covariates References: Desmarais, Bruce and Cranmer, Skyler. "Statistical Inference for Valued-Edge Networks: The Generalized Exponential Random Graph Model" (2012) PLoS ONE Wilson, James, et al. "Stochastic Weighted Graphs: Flexible Model Specification and Simulation" (2015) arxiv Preprint James D. Wilson GERGMs for Weighted Networks 9 / 30

10 Generalized Exponential Random Graph Model Probability measure on family of weighted graphs w/ n nodes, M edges Model: Two Steps 1) Model joint structure of Y on restricted network X [0, 1] M : f X (x, θ) = exp (θ h(x)) [0,1] m exp (θ, x [0, 1]M h(z)) dz h [0, 1] M R p : Network covariates (endogeneous or exogenous) θ R p : unknown structural parameters James D. Wilson GERGMs for Weighted Networks 10 / 30

11 Generalized Exponential Random Graph Model 2) Transform onto space of continuous weights: f Y (y, θ, Λ) = exp (θ h(t (y, Λ))) [0,1] m exp (θ h(z)) dz ij t ij (y, Λ), y R M T R M [0, 1] m : parametric transformation function Monotonically increasing Most obvious choice: cumulative distribution functions Λ R q : unknown transformation parameters James D. Wilson GERGMs for Weighted Networks 11 / 30

12 Features of the GERGM Flexible model for any type of weighted network Like ERGM, requires computationally efficient algorithms for estimation like Markov Chain Monte-Carlo or Maximum pseudo-likelihood Reduces to classical regression when h(x) = 0. James D. Wilson GERGMs for Weighted Networks 12 / 30

13 Model Specification: Weighted Network Statistics Network Statistic Parameter Reciprocity θ R i<j Value x ij x ji Cyclic Triads θ CT (x ij x jk x ki + x ik x kj x ji ) i<j<k In-Two-Stars θ ITS i Out-Two-Stars θ OTS i Edge Density θ E i j α R x ji x ki j<k i x ij x ik j<k i α E x ij α ITS α OTS Transitive Triads θ TT (x ij x jk x ik + x ij x kj x ki + x ij x kj x ik ) + i<j<k α CT i<j<k (x ji x jk x ki + x ji x jk x ik + x ji x kj x ki ) α TT James D. Wilson GERGMs for Weighted Networks 13 / 30

14 Inference on the GERGM Likelihood of GERGM is intractable; relies on MCMC Gibbs (Desmarais, et al., 2012) Major Issue: Restricts model specification by requiring first order network statistics 2 h(x) xij 2 = 0, i, j [n] Metropolis-Hastings (Wilson, et al. 2015): Removes above restriction on GERGM James D. Wilson GERGMs for Weighted Networks 14 / 30

15 Metropolis-Hastings Sampling Framework: Acceptance/Rejection algorithm for weighted edges w/ multivariate truncated normal proposal distribution. Proposal: q σ (w x) = Advantages: σ 1 φ( w x Φ( 1 x σ Flexible model specification Interaction effects σ ) x ) Φ( σ ), 0 w 1 Exponential weighting of covariates New available models can avoid likelihood degeneracy James D. Wilson GERGMs for Weighted Networks 15 / 30

16 GERGM - Metropolis Hastings Procedure 1 For i, j [n], generate proposal edge x (t) independently across edges. 2 Set i,j q σ ( x (t) i,j ) where x (t+1) x (t) = ( x (t) = i,j ) i,j [n] w.p. ρ(x (t), x (t) ) x (t) w.p. 1 ρ(x (t), x (t) ) ρ(x, y) = min ( f m X (y θ) q σ (x i y i ) f X (x θ) i=1 q σ (y i x i ), 1) = min (exp (θ m q σ (x (h(y) h(x))) i y i ), 1) (1) q σ (y i x i ) i=1 James D. Wilson GERGMs for Weighted Networks 16 / 30

17 Case Study: In-Two-Stars Model Model: f X (x, θ, α) = exp (θ Eh E (x) + θ ITS h ITS (x)), x [0, 1] m C(θ E, θ ITS ) Edge density: h E (x) = i j x ij /m In-Two-Stars: h ITS (x, α) = ( i j<k i x ji x ki ) α When α = 1, this is closely related to the well-known degenerate triangle model and the Ising model. James D. Wilson GERGMs for Weighted Networks 17 / 30

18 Case Study: In-Two-Stars Model Degeneracy Issues: In α = 1 model, simulated networks tend to be nearly empty or nearly complete leading to likelihood degeneracy: James D. Wilson GERGMs for Weighted Networks 18 / 30

19 Case Study: In-Two-Stars Model Upshot: Exponential weighting of network statistics alleviates degeneracy! James D. Wilson GERGMs for Weighted Networks 19 / 30

20 Application: International Lending Networks Lending volume (in millions of US $) between 18 countries Collected by Bank of International Settlements (BIS) Originally published in Oatley et al. (2013) James D. Wilson GERGMs for Weighted Networks 20 / 30

21 Heavy Tailed Lending volumes Log normalized volume for edge weights James D. Wilson GERGMs for Weighted Networks 21 / 30

22 Dynamic Summary of Covariates Highly transitive, not reciprocal Temporal trends suggest a change in lending structure James D. Wilson GERGMs for Weighted Networks 22 / 30

23 Goodness of Fit James D. Wilson GERGMs for Weighted Networks 23 / 30

24 Application: U.S. Migration Data -Desmarais et al Describes the inter-state migration in U.S. from 2006 to Fit a GERGM with 5 network statistics, 11 demographic covariates James D. Wilson GERGMs for Weighted Networks 24 / 30

25 Covariate Estimates James D. Wilson GERGMs for Weighted Networks 25 / 30

26 Inference on the Migration Data Statistic in Migration Transitivity Out-two-stars In-two-stars Cyclic triads Unemployment Send. Population Rec. January Temp. Send. Unemployment, population, and average January temperature each influence U.S. migration! James D. Wilson GERGMs for Weighted Networks 26 / 30

27 Goodness of Fit M-H, Gibbs, and MPLE have good performance James D. Wilson GERGMs for Weighted Networks 27 / 30

28 gergm.r Software Tutorial We ll now go through two examples of how to use the GERGM in practice. Go to under "Presentations" to get started. James D. Wilson GERGMs for Weighted Networks 28 / 30

29 Future Work Model Selection for random graph models Mixing Time Analysis Temporal Extensions New applications (always looking for exciting data) James D. Wilson GERGMs for Weighted Networks 29 / 30

30 Thank you Collaborators: Skyler Cranmer, Bruce Desmarais, Matthew Denny, and Shankar Bhamidi. Funding: National Science Foundation Grants: DMS , DMS , SES , SES , SES , and CISE Contact: Place: University of San Francisco, Department of Mathematics James D. Wilson GERGMs for Weighted Networks 30 / 30