# Modeling Transitivity in Local Structure Graph Models

## Abstract

Local Structure Graph Models (LSGMs) describe network data by modeling, and thereby controlling, the local structure of networks in a direct and interpretable manner. Specification of such models requires identifying three factors: a saturated, or maximally possible, graph; a neighborhood structure of dependent potential edges; and, lastly, a model form prescribed by full conditional binary distributions with appropriate “centering” steps and dependence parameters. This last aspect particularly distinguishes LSGMs from other model formulations for network data. In this article, we explore the expanded LSGM structure to incorporate dependencies among edges that form potential triangles, thus explicitly representing transitivity in the conditional probabilities that govern edge realization. Two networks previously examined in the literature, the Faux Mesa High friendship network and the 2000 college football network, are analyzed with such models, with a focus on assessing the manner in which terms reflecting two-way and three-way dependencies among potential edges influence the data structures generated by models that incorporate them. One conclusion reached is that explicit modeling of three-way dependencies is not always needed to reflect the observed level of transitivity in an actual graph. Another conclusion is that understanding the manner in which a model represents a given problem is enhanced by examining several aspects of model structure, not just the number of some particular topological structure generated by a fitted model.

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## Notes

1. The binary conditionals with centered parameterizations in (3.3)–(3.5) induce a curved exponential joint distribution for the LSGMs here and simulations from models fit by pseudo-likelihood (or maximum likelihood) estimation may not produce proportions of realized edges that match, on average, those from the original data. Due to the curved exponential form, the natural parameter space of these LSGMs is also not of the same dimension as the true parameter space; for the Faux Mesa High network, Models 1,2,3 have 4,4,5 parameters, respectively, while the dimensions of a full rank minimally sufficient statistic are 7,10,14 in these models. See the Supplementary Materials for more details on these joint distributions and minimal sufficiency.

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Correspondence to Emily Casleton.

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Casleton, E., Nordman, D.J. & Kaiser, M.S. Modeling Transitivity in Local Structure Graph Models. Sankhya A 84, 389–417 (2022). https://doi.org/10.1007/s13171-021-00264-1

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• DOI: https://doi.org/10.1007/s13171-021-00264-1

### Keywords

• Conditionally specified models
• network analysis
• network model assessment
• random graphs
• transitivity.

### AMS (2000) subject classification

• Primary 05C80
• Secondary 62M05