Statistical network modeling: challenges and perspectives

Transcription

1 Statistical network modeling: challenges and perspectives Harry Crane Department of Statistics Rutgers August 1, 2017 Harry Crane (Rutgers) Network modeling JSM IOL / 25

2 Statistical network modeling: challenges and perspectives Harry Crane Department of Statistics Rutgers August 1, 2017 Subject of upcoming book on network modeling (Summer 2018) Based on joint work with Walter Dempsey (Harvard) Will speak Thursday, August 4 at 11:35 AM in Room CC-309. Harry Crane (Rutgers) Network modeling JSM IOL / 25

3 Modeling network data: What is it? Where does it come from? Harry Crane (Rutgers) Network modeling JSM IOL / 25

4 Modeling network data: What is it? Where does it come from? Assumed Setting: Population Observed network (sample) Guiding Question: How to model network data in the presence of sampling? Harry Crane (Rutgers) Network modeling JSM IOL / 25

5 Key Points and Further Reading Important directions of future statistics research: 1 Network sampling 2 Network dynamics mostly ignored in the current statistics literature References: H. Crane and W. Dempsey. (2015). A framework for statistical network modeling. arxiv: H. Crane and W. Dempsey. (2016). Edge exchangeable models for interaction networks. Journal of the American Statistical Association. H. Crane. Principles of Statistical Network Modeling (tentative title). Forthcoming, H. Crane. (2017). Exchangeable graph-valued Feller processes. Probability Theory and Related Fields. H. Crane. (2017). Combinatorial Lévy processes. Annals of Applied Probability. Available at Harry Crane (Rutgers) Network modeling JSM IOL / 25

6 Network modeling Conventional Definition: A (parameterized) statistical model is a family of probability distributions M = {P θ : θ Θ}, each defined on the sample space. Population or Sample model? And what s the connection? Population Observed network (sample)??? Model {P θ : θ Θ}??? Problem: How to draw sound inferences about population model based on sampled network? Need to model data in a manner consistent with (i) population model and (ii) sampling mechanism. Harry Crane (Rutgers) Network modeling JSM IOL / 25

7 Network modeling Revised Definition: A (parameterized) statistical model is (i) a family of probability distributions M = {P θ : θ Θ}, each defined on the sample space, along with (ii) a description of the sampling scheme (S n,n ) 1 n N used in observing a sample of each size n. Population Observed network (sample) Y N S n,n (Y N ) Model {P θ : θ Θ}??? Sampling scheme S m,n necessary to establish relationship between observation and population. Sampling mechanism often (almost always) left out of model specification. Harry Crane (Rutgers) Network modeling JSM IOL / 25

8 Preview Will discuss 3 network modeling scenarios: 1 Scenario 1: Vertex sampling ERGMs, graphon models, Aldous Hoover theorem, vertex exchangeability 2 Scenario 2: Edge sampling edge exchangeability, Hollywood model 3 Scenario 3: Time-varying networks Graph-valued Markov chains Modeling network dynamics under sampling Harry Crane (Rutgers) Network modeling JSM IOL / 25

9 Scenario 1: Inference from sampled networks Only n of all N Facebook account users are sampled and their relationships to one another are recorded as an array Y n = (Y ij ) 1 i,j n with { 1, i and j Facebook friends, Y ij = 0, otherwise. We are interested in using information Y n to infer structure of entire Facebook network. Population Y N Sample Y n How should Y n be modeled if we are interested in inferring the structure of relationships among all students in the population Y N? Depends on how Y n was sampled from Y N. Harry Crane (Rutgers) Network modeling JSM IOL / 25

10 Scenario 1: ERGM as population model Given any sufficient statistics (T 1,..., T k ) and parameters (θ 1,..., θ k ), assign probability { k } Pr(Y = y; θ, T ) exp θ i T i (y), y = (y ij ) 1 i,j N {0, 1} N N. i=1 Holland and Leinhardt (1981), Frank and Strauss (1986), Wasserman and Pattison (1996), Wasserman and Faust (1994). Typical approach: Estimate θ by fitting ERGM (θ) to Y n, obtain ˆθ n and use as estimate for θ in population. Population Y N Sample Y n Model ERGM (θ)??? Parameter θ θ Harry Crane (Rutgers) Network modeling JSM IOL / 25

11 Consistency under subsampling for ERGM Theorem (Shalizi Rinaldo) Model for S n,n (Y n ) is ERGM (θ) if and only if sufficient statistics T have separable increments. separable increments: in-degree, out-degree, reciprocity non-separable increments: transitivity (triangles), most higher order statistics Unclear how to proceed with inference for θ based on sample. Population Y N Sample Y n Model ERGM (θ)??? Parameter θ θ Estimate??? ˆθ n Harry Crane (Rutgers) Network modeling JSM IOL / 25

12 Graphon models (Glorified Erdős Rényi model) Originally appeared as φ-processes in Diaconis Freedman (1980), also Aldous (1983). Graphon φ : [0, 1] [0, 1] [0, 1] Associate i.i.d. sequence U 1, U 2,... of Uniform[0, 1] random variables to vertices 1, 2,.... Construct Y N = (Y ij ) 1 i,j N by conditionally independently for all 1 i, j N. Pr(Y ij = 1 U 1, U 2,...) = φ(u i, U j ) (1) Population Y N Sample Y n Model graphon (φ) graphon (φ) Parameter φ φ Estimate ˆφ n ˆφ n Harry Crane (Rutgers) Network modeling JSM IOL / 25

13 Implications of Graphon models Vertex exchangeable: Assign same probability to Theorem (Aldous Hoover) Every vertex exchangeable model for countable population is a mixture of graphon models. Vertex-exchangeability sampled vertices representative of the population of all vertices not realistic in most applications. Cannot explain sparse behavior. Graphon models achieve consistency under subsampling, but vertex exchangeability implies unrealistic sampling scheme and network properties. Harry Crane (Rutgers) Network modeling JSM IOL / 25

14 Scenario 2: Phone calls from a database Entries are sampled uniformly at random from a large database of phone calls (or s). Each observation (C i, R i ) contains identity of the caller C i and receiver R i on the ith sampled call. Interested in inferring the structure of connections among users in the database. Caller Receiver Time of Call (a) (b) 15: (c) (a) 15: (d) (e) 16: (c) (a) 15: Call sequence X 1 = (a, b), X 2 = (c, a), X 3 = (d, e), X 4 = (a, c) induces network:.... Harry Crane (Rutgers) Network modeling JSM IOL / 25

15 Interaction Networks Dataset vertices edges Actor collaborations actors movies Enron corpus employees s Karate club dataset club members social interactions Wikipedia voting Wikipedia admin. votes US Airport airports flights Scientific collaborations scientists articles UC Irvine online community members online messages Political blogs Websites hyperlinks These datasets are driven by interactions Edges are the units not represented as a (vertex-labeled) graph Harry Crane (Rutgers) Network modeling JSM IOL / 25

16 Edge exchangeable models Phone calls are sampled uniformly from the database exchangeable sequence of pairs (C 1, R 1 ), (C 2, R 2 ),.... Assign same probability to Edge exchangeability Size-biased vertex sampling Hollywood model: Easy for estimation, prediction, and testing questions. Sparse with probability 1 for 1/2 < α < 1. Power law with exponent α + 1 for 0 < α < 1. H. Crane and W. Dempsey. (2016). Edge exchangeable models for interaction networks. Journal of the American Statistical Association, to appear. Harry Crane (Rutgers) Network modeling JSM IOL / 25

17 Scenario 3: Dynamics in social networks For each day t = 0, 1, 2,... an array Y(t) = (Y ij (t)) 1 i,j N records interactions among a sample of Twitter users with { 1, i retweeted or liked a tweet by j, Y ij (t) = 0, otherwise. Interested in modeling how past interaction behavior may be indicative of future behavior for the population of all Twitter users. What kinds of Markov models can be fit to this data? Harry Crane (Rutgers) Network modeling JSM IOL / 25

18 Scenario 3: Dynamics in social networks For each day t = 0, 1, 2,... an array Y(t) = (Y ij (t)) 1 i,j N records interactions among a sample of Twitter users with { 1, i retweeted or liked a tweet by j, Y ij (t) = 0, otherwise. Interested in modeling how past interaction behavior may be indicative of future behavior for the population of all Twitter users. What kinds of Markov models can be fit to this data? Harry Crane (Rutgers) Network modeling JSM IOL / 25

19 Illustration: TERGM θ: parameter vector η(θ): natural parameter g(g, G ): sufficient statistic for transition from G to G Observations: P{Γ t+1 = G Γ t = G; θ} exp{η(θ) g(g, G )}. Typically choose g so that g(g, G ) = g(g σ, G σ ) for arbitrary permutations σ : [n] [n]. Not projective in general (Shalizi & Rinaldo, 2013) Robins and Pattison (2001), Hanneke, Fu and Xing (2010), Kravitsky & Handcock (2012) What do exchangeability and projective Markov properties imply about transition behavior? Harry Crane (Rutgers) Network modeling JSM IOL / 25

20 Time-varying network models Basic assumptions: (Y(t)) t 0 is a Markov process satisfying exchangeability: for all σ : N N, (Y σ (t)) t 0 has same finite-dimensional distributions as Y. projective Markov property: (Y(t) [n] ) t 0 is a Markov chain for every n = 1, 2,.... Harry Crane (Rutgers) Network modeling JSM IOL / 25

21 Time-varying network models Basic assumptions: (Y(t)) t 0 is a Markov process satisfying exchangeability: for all σ : N N, (Y σ (t)) t 0 has same finite-dimensional distributions as Y. projective Markov property: (Y(t) [n] ) t 0 is a Markov chain for every n = 1, 2,.... Harry Crane (Rutgers) Network modeling JSM IOL / 25

22 Time-varying network models Basic assumptions: (Y(t)) t 0 is a Markov process satisfying exchangeability: for all σ : N N, (Y σ (t)) t 0 has same finite-dimensional distributions as Y. projective Markov property: (Y(t) [n] ) t 0 is a Markov chain for every n = 1, 2,.... Harry Crane (Rutgers) Network modeling JSM IOL / 25

23 Time-varying network models Basic assumptions: (Y(t)) t 0 is a Markov process satisfying exchangeability: for all σ : N N, (Y σ (t)) t 0 has same finite-dimensional distributions as Y. projective Markov property: (Y(t) [n] ) t 0 is a Markov chain for every n = 1, 2,.... Harry Crane (Rutgers) Network modeling JSM IOL / 25

24 Rewiring processes (Crane, 2015, 2017) Let φ : [0, 1] [0, 1] [0, 1] [0, 1] be a Markovian graphon : Write φ(u, v) = (φ 0 (u, v), φ 1 (u, v)) for (u, v) [0, 1] [0, 1]. Rewiring process: Make a transition G G by generating a transition probability for each pair of vertices: ( ) 1 φ0 (U i, U j ) φ 0 (U i, U j ) 1 φ 1 (U i, U j ) φ 1 (U i, U j ) Rewire every edge according to transition probability determined by φ(u i, U j ). Theorem (Crane (2017)) Every exchangeable, projective Markov chain is a rewiring process. H. Crane. (2015). Time-varying network models. Bernoulli. H. Crane. (2017). Exchangeable graph-valued Feller processes. Probability Theory and Related Fields. H. Crane. (2017+). Combinatorial Lévy processes. Annals of Applied Probability. Harry Crane (Rutgers) Network modeling JSM IOL / 25

25 High Level Takeaways sampling statistical units network rep. model selection vertices graph ERGM edge sampling edges edge-labeled network SBM snowball ego-network graphon traceroute. paths. 1 Sampling scheme a key part of modeling. Sampling scheme often depends on the network itself. 2 Edge sampling, snowball sampling, traceroute sampling all depends on network Sampling affects invariance principles of model affects inference. 3 Vertex exchangeability, edge exchangeability, relative exchangeability, relational exchangeability How are network dynamics affected by sampling? Wide open.. Harry Crane (Rutgers) Network modeling JSM IOL / 25

26 Further reading Network Analysis: E. Kolaczyk. (2009). Statistical Analysis of Network Data: Methods and Models. Network Modeling: H. Crane and W. Dempsey. (2015). A framework for statistical network modeling. arxiv: H. Crane. Principles of Statistical Network Modeling (tentative title). Forthcoming, Goldenberg, Zheng, Fienberg and Airoldi. (2009). A survey of statistical network models. arxiv: Exchangeability: D.J. Aldous. (1983). Exchangeability and Related Topics. H. Crane and W. Dempsey. (2016). Edge exchangeable models for interaction networks. Journal of the American Statistical Association. H. Crane and H. Towsner. (2015). Relative exchangeability. Dynamic networks: H. Crane. (2015). Time-varying network models. Bernoulli. H. Crane. (2017). Exchangeable graph-valued Feller processes. Probability Theory and Related Fields. H. Crane. (2017+). Combinatorial Lévy processes. Annals of Applied Probability. Harry Crane (Rutgers) Network modeling JSM IOL / 25