Graver basis for an undirected graph and its application to testing the beta model of random graphs

Ogawa, Mitsunori; Hara, Hisayuki; Takemura, Akimichi

doi:10.1007/s10463-012-0367-8

Cited by 17 publications

(26 citation statements)

References 27 publications

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“…A similar problematic has been observed for other models, for instance for network models (Petrović et al, ). Here, the largest models considered so far are food webs of around 40 nodes (Gross et al, ; Ogawa et al, ). Kahle & Rauh () have a collection of Markov bases that have been computed during the last years.…”

Section: Introductionmentioning

confidence: 99%

Exact Goodness‐of‐Fit Testing for the Ising Model

Campo

Cepeda

Uhler

2016

Scandinavian J Statistics

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Abstract. The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness-of-fit test for the finite-lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness-of-fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness-of-fit testing for the Ising model which avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.

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Section: Introductionmentioning

confidence: 99%

Exact Goodness‐of‐Fit Testing for the Ising Model

Campo

Cepeda

Uhler

2016

Scandinavian J Statistics

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“…The FIM in (21) can now be obtained from (35). Under the assumptions preceding (22), it follows that…”

Section: Discussionmentioning

confidence: 99%

“…Goodness-of-fit tests are omitted from the discussion, and we instead refer the reader to earlier studies on ERGMs [34], the undirected β-model [35], and the directed β-model [9], [36].…”

Section: The Generalized β-Modelmentioning

confidence: 99%

The $\beta$-Model—Maximum Likelihood, Cramér–Rao Bounds, and Hypothesis Testing

Wahlström¹,

Skog²,

Rosa³

et al. 2017

IEEE Trans. Signal Process.

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Abstract-We develop a maximum-likelihood based method for regression in a setting where the dependent variable is a random graph and covariates are available on a graph-level. The model generalizes the well-known β-model for random graphs by replacing the constant model parameters with regression functions. Cramér-Rao bounds are derived for the undirected β-model, the directed β-model, and the generalized β-model. The corresponding maximum likelihood estimators are compared to the bounds by means of simulations. Moreover, examples are given on how to use the presented maximum likelihood estimators to test for directionality and significance. Last, the applicability of the model is demonstrated using dynamic social network data describing communication among healthcare workers.

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“…Notable examples include works such as [SS06, Theorem 4] on toric geometry of compatible full conditionals, [Mor13, Theorem 3.2] on relations among conditional probabilities, [SW12] that studies algebra and geometry of statistical ranking models and the ascending model in particular, and [PRF10] that study the structure of Markov and Graver bases for the directed model p 1 whose restriction (to the part of the random graph consisting only of reciprocated edges) is the β model. More recently, [OHT13] offer the exact test implementation whose main ingredient is a random generation of Markov and Graver bases elements for the β model.…”

Section: Sampling Algorithms In Exact Testingmentioning

confidence: 99%

“…This allows the authors to construct a swap-free algorithm for sampling the restricted fiber of the β model. If one is interested in sampling a restricted fiber for any general linear ERGM or any log-linear model, [OHT13] and [AHT12] show that one needs a much larger set of moves to guarantee connectivity, i.e. minimal Markov bases may not do the job.…”

Section: Sampling Algorithms In Exact Testingmentioning

confidence: 99%

A survey of discrete methods in (algebraic) statistics for networks

Petrović¹

2017

Algebraic and Geometric Methods in Discrete Mathematics

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Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view of a particular family of statistical models for networks called exponential random graph models. The problems and underlying constructions are also related to well-known concepts in commutative algebra and graph-theoretic concepts in computer science. We outline a few lines of recent work that highlight the natural connection between these fields and unify them into some open problems. While these problems are often relevant in discrete mathematics in their own right, the emphasis here is on statistical relevance with the hope that these lines of research do not remain disjoint. Suggested specific open problems and general research questions should advance algebraic statistics theory as well as applied statistical tools for rigorous statistical analysis of networks.

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Graver basis for an undirected graph and its application to testing the beta model of random graphs

Abstract: In this paper we give an explicit and algorithmic description of Graver basis for the toric ideal associated with a simple undirected graph and apply the basis for testing the beta model of random graphs by Markov chain Monte Carlo method.

Cited by 17 publications

References 27 publications

Exact Goodness‐of‐Fit Testing for the Ising Model

Exact Goodness‐of‐Fit Testing for the Ising Model

The $\beta$-Model—Maximum Likelihood, Cramér–Rao Bounds, and Hypothesis Testing

A survey of discrete methods in (algebraic) statistics for networks

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