Algebraic statistics for network models

Transcription

1 Algebraic statistics for network models Connecting statistics, combinatorics, and computational algebra Part One Sonja Petrović (Statistics Department, Pennsylvania State University) Applied Mathematics Department, Illinois Institute of Technology Summer School on Network Science Columbia, SC Monday, 20 May 2013 S. Petrović Algebraic Statistics for Network Models Monday, 20 May / 22

2 General framework and motivation Network analysis has advanced - Empirical work, simple models, probabilistic properties. - But: surprising non-standard properties, new theoretical challenges in statistics Some approaches that have a statistical grounding do not necessarily scale well to large sparse network settings - ( model/data fit ) - Recent: degenerate statistical behavior of network modeling tools Motivation: practical problems for network data structures where the number of variables, parameters is large (relative to the number of independent observations). Algebraic statistics: Insights to a variety of categorical data problems insights aid model development and analysis processes S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

3 What is Algebraic Statistics? Algebraic geometry and related fields applied to statistics Fact (Guiding principle) Many important statistical models correspond to algebraic or semi-algebraic sets of parameters. The geometry of these parameter spaces determines the behavior of widely used statistical inference procedures. S. Petrović Algebraic Statistics for Network Models Monday, 20 May / 22

4 What is Algebraic Statistics? Algebraic geometry and related fields applied to statistics Fact (Guiding principle) Many important statistical models correspond to algebraic or semi-algebraic sets of parameters. The geometry of these parameter spaces determines the behavior of widely used statistical inference procedures. Model geometry: Shape of a statistical model: intuitive notion of fundamental importance to statistical inference; reflected in its abstract geometric properties Ex: is the likelihood function multimodal? Does the model have singularities (is non-regular)? Nature of underlying singularities? When a model is algebraic, use tools from algebraic geometry and computational algebra software packages. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

5 Motivating problem Model Validation Problem Tools: algebra and polyhedral geometry Problem Given a candidate ERGM P and one observed network x, decide (with a Markov bases high degree of confidence) whether x can be regarded as a draw from Markov bases some distribution P θ0 P. Maximum likelihood estimation problem: Use the observed data x to produce an optimal estimate for θ 0. JavaView v Goodness-of-fit problem (and model selection): Can the MLE be considered as a satisfactory generative model for the data at hand? Loading Sonja Petrović (SAC seminar) Algebraic statistics Februa S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

6 Motivating problem Model Validation Problem Tools: algebra and polyhedral geometry Problem Given a candidate ERGM P and one observed network x, decide (with a high degree of confidence) whether x can be regarded as a draw from Markov bases some distribution P θ0 P. Markov bases Maximum likelihood estimation problem: Use the observed data x to produce an optimal estimate for θ 0. (Faces of model polytope.) JavaView v Loading Goodness-of-fit problem (and model selection): Can the MLE be considered as a satisfactory generative model for the data at hand? Sonja Petrović (SAC seminar) Algebraic statistics Februa S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

7 Motivating problem Model Validation Problem Tools: algebra and polyhedral geometry Problem Given a candidate ERGM P and one observed network x, decide (with a high degree of confidence) whether x can be regarded as a draw from Markov bases some distribution P θ0 P. Markov bases Maximum likelihood estimation problem: Use the observed data x to produce an optimal estimate for θ 0. (Faces of model polytope.) JavaView v Goodness-of-fit problem (and model selection): Can the MLE be considered as a satisfactory generative model for the data at hand? (Markov bases for random walk on a fiber.) Loading Sonja Petrović (SAC seminar) Algebraic statistics Februa S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

8 Statistical network analysis What is a model? Statistical Network (Random Graph) Analysis Let G n be the set of simple graphs on n nodes: G n = 2 (n 2). * Nodes = units of some (sub)population of interest. * Edges = a set of static relationships among units. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

9 Statistical network analysis What is a model? Statistical Network (Random Graph) Analysis Let G n be the set of simple graphs on n nodes: G n = 2 (n 2). * Nodes = units of some (sub)population of interest. * Edges = a set of static relationships among units. - Reprensentation: 0/1 adjacency matrix (n n), or a point in {0, 1} n ; OR: a ( n 2) -dimensional 0/1 vector indexed by node pairs. Example Graph with an edge {1, 2} and a triple edge {1, 3}: S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

10 Statistical network analysis What is a model? Statistical Network (Random Graph) Analysis Let G n be the set of simple graphs on n nodes: G n = 2 (n 2). * Nodes = units of some (sub)population of interest. * Edges = a set of static relationships among units. - Reprensentation: 0/1 adjacency matrix (n n), or a point in {0, 1} n ; OR: a ( n 2) -dimensional 0/1 vector indexed by node pairs. Example Graph with an edge {1, 2} and a triple edge {1, 3}: x = [1, 3, 0,..., 0] T. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

11 Statistical network analysis What is a model? Statistical Network (Random Graph) Analysis Let G n be the set of simple graphs on n nodes: G n = 2 (n 2). * Nodes = units of some (sub)population of interest. * Edges = a set of static relationships among units. - Reprensentation: 0/1 adjacency matrix (n n), or a point in {0, 1} n ; OR: a ( n 2) -dimensional 0/1 vector indexed by node pairs. Example Graph with an edge {1, 2} and a triple edge {1, 3}: x = [1, 3, 0,..., 0] T. A statistical model is a collection of probability distributions over graphs indexed by a set of parameters. The form and properties of such statistical models are dictated by the goals of the analysis at hand. With a model in place, we can determine the probability of any network topology. We can also estimate the model parameters that best fit given data. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

12 Statistical network analysis Why search for a good model? Statistical network analysis - what it can tell us Why search for a well-fitting statistical model of an observed social network?s Allows us to understand the uncertainty associated with observed outcomes. Allows inferences about whether network substructures are more commonly observed than by chance. Allows for simulation. Allows for the assessment of local effects (reciprocation, attractiveness, desire to expand, etc). Statistical models for networks: Classes of probability distributions for graphs, interpretable, realistic models for large distributions based on edges (modeling the random occurrence of edges). S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

13 ERGMs A well-known family of statistical models Exponential Random Graph (ERG) Models Specify a set of informative network statistics on G n (capture key features of the network) t : G n R d, x t(x) = (t 1 (x),..., t d (x)) R d, such that the probability of observing x is a function of t(x) only. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

14 ERGMs A well-known family of statistical models Exponential Random Graph (ERG) Models Specify a set of informative network statistics on G n (capture key features of the network) t : G n R d, x t(x) = (t 1 (x),..., t d (x)) R d, such that the probability of observing x is a function of t(x) only. Standard examples: the number of edges E(x) (Erdös-Renyi model); number of triangles T (x); the number of k-stars; S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

15 ERGMs A well-known family of statistical models Exponential Random Graph (ERG) Models Specify a set of informative network statistics on G n (capture key features of the network) t : G n R d, x t(x) = (t 1 (x),..., t d (x)) R d, such that the probability of observing x is a function of t(x) only. Standard examples: the number of edges E(x) (Erdös-Renyi model); number of triangles T (x); the number of k-stars; More elaborate examples (number of statistics grows with n) the degree sequence: β-model [RPF 2011; Chatterjee-Diaconis 2011] in- and out-degrees (directed): p 1 model [Holland-Leinhardt 1981] T (x) and k-stars: Markov graph model [Frank-Strauss 1986] (log-linear models over a set of edges) S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

16 ERGMs may have non-standard properties (e.g. increasing number of parameters) Number of parameters increasing with n In general, it is desirable to have an increasing number of parameters in order to provide more descriptive and flexible models. However, the number of parameters can only grow at a o(n) rate, otherwise inference is not possible. Examples of network models in which the number of parameters grow with the size n of the network are the β-model; the Markov random graph models; models related to JDM (Aaron Dutle, Wed 5/22); the SBM with growing number of hidden communities. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

17 ERGMs may have non-standard properties But: significant dimension reduction property (well-understood) Sufficiency and graph equivalence classes ERGMs are models over equivalence classes of G n, where two graphs x and y are regarded probabilistically equivalent whenever t(x) = t(y). ET Model: we consider G 9 with network statistics (E(x), T (x)). There are 2 36 distinct graphs but only 444 distinct network statistics Number of triangles Number of edges β-model: for n = 6, 7, 8, 9, there are 6944, 11850, , distinct (ordered) degree sequences (Stanley, 1991). S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

18 A simplest ERGM for random graphs- -with sufficient statistics of interest to many communities! The β model for random graphs: degree sequences The β-model is the ERGM on labeled networks with network statistics given by the (ordered) degree sequence: x G n d(x) = d = (d 1,..., d n ) N n, where d i is the degree of node i. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

19 A simplest ERGM for random graphs- -with sufficient statistics of interest to many communities! The β model for random graphs: degree sequences The β-model is the ERGM on labeled networks with network statistics given by the (ordered) degree sequence: x G n d(x) = d = (d 1,..., d n ) N n, where d i is the degree of node i. Definition (Set of all possible outcomes for generalized Beta) S n := {x i,j : i < j and x i,j {0, 1,..., N i,j }} N (n 2). Definition (Parametrization of the beta model) For β R n : p i,j = eβ i +β j 1 + e β i +β j and p j,i = 1 p i,j = e β i +β j, i j. The model M β for n vertices consists of all p i,j s of this form. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

20 What parameters best explain the given (network) data? MLE non-existence leads to degeneracy Inference Problem Given one observation x G n (given t = t(x)), estimate the parameters. In the ET model, want to learn 2 parameters, in the β-model, n. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

21 What parameters best explain the given (network) data? MLE non-existence leads to degeneracy Inference Problem Given one observation x G n (given t = t(x)), estimate the parameters. In the ET model, want to learn 2 parameters, in the β-model, n. MLE(p) := argmax p Mn i<j p x ij ij. Fact: The MLE is nonexistent if ˆp is on the boundary of M A. Consequence: Some of the coordinates of ˆp are either 0 or 1, and are, therefore, non-estimable. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

22 What parameters best explain the given (network) data? MLE non-existence leads to degeneracy Inference Problem Given one observation x G n (given t = t(x)), estimate the parameters. In the ET model, want to learn 2 parameters, in the β-model, n. MLE(p) := argmax p Mn i<j p x ij ij. Fact: The MLE is nonexistent if ˆp is on the boundary of M A. Consequence: Some of the coordinates of ˆp are either 0 or 1, and are, therefore, non-estimable. Key tasks (Fundamental for goodness-of-fit testing) (1) decide whether the MLE exists (2) identify non-estimable parameters. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

23 This crucial problem must not be overlooked. MLE non-existence leads to degeneracy A small example of data leading to a nonexistent MLE 0 N 1,2 N 3,4 0 (The model: p i,j = eβ i +β j 1+e β i +β j.) Left: data exhibiting the above pattern, when N i,j = 3 for all i j. Right: table of the extended MLE of the estimated probabilities. Under natural parametrization, the supremum of the log-likelihood is achieved in the limit for any sequence of natural parameters {β (k) } of the form β (k) = ( c k, c k, c k, c k ), where c k as k. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

24 This crucial problem must not be overlooked. MLE non-existence leads to degeneracy A small example of data leading to a nonexistent MLE Problem (!) Current algorithms and software for fitting ERGMs have no simple mechanism for detecting non-existence and identifying non-estimable parameters and degeneracy Left: data exhibiting the above pattern, when N i,j = 3 for all i j. Right: table of the extended MLE of the estimated probabilities. Under natural parametrization, the supremum of the log-likelihood is achieved in the limit for any sequence of natural parameters {β (k) } of the form β (k) = ( c k, c k, c k, c k ), where c k as k. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

25 Geometry of Discrete Exponential Families Polytopes and MLE existence Basics of Discrete Exponential Families The set P = convhull(t(x) : x G n ) is called the model polytope. int(p)= {E θ [t], θ Θ} is precisely the set of all possible expected values of t (mean value space; homeomorphic to parameter space). Theorem The MLE exists for x if and only if t(x) int(p). S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

26 Geometry of Discrete Exponential Families Polytopes and MLE existence Basics of Discrete Exponential Families The set P = convhull(t(x) : x G n ) is called the model polytope. int(p)= {E θ [t], θ Θ} is precisely the set of all possible expected values of t (mean value space; homeomorphic to parameter space). Theorem The MLE exists for x if and only if t(x) int(p). 90 Convex support In the ET example, MLE exists for 415 of 444 cases. The boundary of P specifies degenerate distributions for the nonexistent MLEs:extended exponential family. Number of triangles Number of edges S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

27 Do we understand the geometry of the β model? Model polytope for β What is its polytope? It is parametrized by the vertex-edge incidence matrix of a complete graph: A 4 = rows indexed by the vertices; columns indexed by (i, j) with i < j. Definition (The model polytope) Example S n := conv {A n x, x S n } Represent the graph with an edge {1, 2} and a triple edge {1, 3} as x = [1, 3, 0,..., 0] T S n. Corresponding point in the model polytope is A n x = [4, 1, 3, 0,... ]. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

28 Do we understand the geometry of the β model? The polytope of degree sequences What is its polytope? Definition The polytope of degree sequences is P n := convhull ({Ax, x G n }). Facet-defining inequalities of P n are known (Mahadev-Peled 96). Theorem (Rinaldo-P.-Fienberg) Let x S n be the observed vector of edge counts. The MLE exists if and only if x j,i + x i,j int(p n ), i = 1,..., n. N i,j N i,j j<i j>i Example (Stanley) f (P 8 ) = (334982, , , , , , 81144, 3322, 1). We used polymake for the computations on small polytopes. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

29 Model polytope for the β model? Combinatorics and facial sets Facial sets of the model polytope Proposition (Rinaldo-P.-Fienberg) A point y belongs to the interior of some face F of P n if and only if there exists a set F {(i, j), i < j} such that for any p = {p i,j : i < j, p i,j [0, 1]} satisfying y = A n p, p i,j {0, 1} if (i, j) F and p i,j (0, 1) if (i, j) F. F is called a facial set of S n, and F c a co-facial set. The MLE does not exist for the graph x if and only if the set {(i, j): i < j, x i,j = 0 or N i,j } contains a co-facial set. Facial sets specify which probability parameters are estimable: only the probabilities {p i,j, (i, j) F}. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

30 Model polytope for the β model? Combinatorics and facial sets Example: Co-facial sets for nonexistent MLEs 0 N 1,2 N 3,4 0 0 N 1,2 0 0 N 3,2 N 4, N 1,2 N 1,3 N 4,1 0 0 N 1,2 0 N 1,3 N 2,3 N 1, N 2,3 N 2,4 Table: Co-facial sets for P 4 (empty cells indicate any entry values). S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

31 Geometry of network models Polytopes and general algorithms MLE existence Rinaldo-Petrović-Fienberg, Beta model, Annals of Statistics 2013 Non-existence of MLEs occurs commonly in large sparse network models, but most often goes undetected. Methodologies for estimation and model validation under a non-existent MLE with proven statistical performance have yet to be developed. Techniques from (polyhedral) geometry offer: The only way to detect non-existence An exact handle on what parameters of the model are actually estimable An algorithmic approach to how to estimate such parameters. Algorithms Related algorithms for finding the facial sets of the model cone are described in Fienberg-Rinaldo 12. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

32 Geometry of network models Overview of Algorithms Polytopes and general algorithms Two tasks: Decide if t(x) P n : existence of the MLE (easy) Decide for which face F of P n, t relint(f ): identification of estimable parameters (hard). For network models, we propose a 2-step procedure 1 Cayley trick (or lifting): replace P n with a larger set which is simpler to analyze: C n, a polyhedral cone. Cayley trick: details 2 Find F using the boundary of C n. Cone Boundary: details S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

33 This is a general framework that applies to any toric model Two related toric models where main Theorem applies Definition (Random graphs with fixed degree sequence) In the special case when N ij = 1, the support S n reduces to G n := {0, 1} (n 2), undirected simple graphs on n nodes. Corollary (RPF) A conjecture in Chatterjee-Diaconis-Sly ( 10) is true: for the random graph model, the MLE exists if and only if d(x) int P n. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

34 This is a general framework that applies to any toric model Two related toric models where main Theorem applies Definition (Random graphs with fixed degree sequence) In the special case when N ij = 1, the support S n reduces to G n := {0, 1} (n 2), undirected simple graphs on n nodes. Corollary (RPF) A conjecture in Chatterjee-Diaconis-Sly ( 10) is true: for the random graph model, the MLE exists if and only if d(x) int P n. Definition (The Rasch model) A random bipartite graph model, the support being G k,l, the set of bipartite graphs on k and l vertices. Theorem (RPF) The MLE of the Rasch model parameters exists if and only if d(x) int P p,q, the polytope of bipartite degree sequences. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

35 This is a general framework that applies to any toric model Extensions (1) Removing the sampling constraint: Let quantities N i,j be random! Theorem (Thanks to Haase and Yu) The model polytope has 3n facets, and is obtained from the product of simplices by removing the vertices {e i e i }, i = 1,..., n. (2) Specialize (1) to directed graphs without multiple edges. This is the Bradley-Terry model for pairwise comparisons. Theorem (Zermelo 29, Ford 57) If the graph is strongly connected, then the MLE exists. Algorithms for detecting co-facial sets still apply. The matrix of the model polytope has dimension ( ( n 2) + n) n(n 1). S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

36 This is a general framework that applies to any toric model Extensions (3) A directed random graph model used in social networking: the p 1 model (Holland-Leinhardt 81). The model polytope is the Minkowski sum of ( n 2) polytopes. Example (n=4) 4 10 = 1, 048, 576 different graphs x. Three cases of the p 1 model: 1 There are 225, 025 points A 4 x, and the MLE exists for 7, , 500, the MLE exists in 12, 684 cases 3 583, 346, the MLE never exists. Theorem (Rinaldo-P.-Fienberg) Sufficient conditions for MLE existence, with large probability, as n grows. In the case of fixed degree sequence graphs, our asymptotic results improve those of Chatterjee-Diaconis 11. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

37 How a network is generated is crucial to properly calculate statistical network properties! HOMEWORK 1 Read section of Social and Economic Networks by Matthew O. Jackson. 2 Think about lattice points of the model polytope for β-model in terms of graphical degree sequences. S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

38 Extra stuff See intro of the beta model annals paper, why the beta model, why the generalization. S. Petrović Algebraic Statistics for Network Models Monday, 20 May / 22

39 Extra slide Extra stuff Characterization of the sets of edges and non-edges in a given graph leading to a nonexistence of the MLE. Examples of a nonexistent MLE (the degrees are non-trivial) for 4, 5, 6 nodes: S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

40 Extra stuff Overview of Algorithms in more detail: Cayley trick The combinatorial complexity of P n make it computationally intractable. For instance, the f -vector of P 8 is (334982, , , , , , 81144, 3322). Cayley Trick. Instead of dealing directly with P n a Minkowski sum of line segments we construct C n, a larger full-dimensional polyhedral cone with ( n 2) more facets than Pn but with many less vertices. Despite its larger dimension, C n is amenable to computational analysis. Theorem Every facial set of P n corresponds to one facial set of C n. Go back to the presentation S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22

41 Extra stuff Overview of Algorithms in more detail: The faces of C n To determine which face F of C n is such that t(x) F is a non-linear optimization problem. We propose two solutions: 1 Repeated linear programming. 2 Non-linear optimization. Go back to the presentation S. Petrović (sonja@psu.edu) Algebraic Statistics for Network Models Monday, 20 May / 22