[Paper Review] Baseline Mixture Models for Social Networks

Carter T. Butts|arXiv (Cornell University)|Oct 8, 2017

Complex Network Analysis Techniques36 references3 citations

TL;DR

This paper introduces baseline mixture models—specifically beta-Bernoulli and Dirichlet-categorical graphs—as flexible, analytically tractable alternatives to standard Bernoulli or uniform graph models for social networks. By modeling network density or dyad frequencies as random variables drawn from continuous mixtures (e.g., beta or Dirichlet distributions), the approach captures cross-graph heterogeneity and enables robust, efficient network inference with fewer observations than traditional methods.

ABSTRACT

Continuous mixtures of distributions are widely employed in the statistical literature as models for phenomena with highly divergent outcomes; in particular, many familiar heavy-tailed distributions arise naturally as mixtures of light-tailed distributions (e.g., Gaussians), and play an important role in applications as diverse as modeling of extreme values and robust inference. In the case of social networks, continuous mixtures of graph distributions can likewise be employed to model social processes with heterogeneous outcomes, or as robust priors for network inference. Here, we introduce some simple families of network models based on continuous mixtures of baseline distributions. While analytically and computationally tractable, these models allow more flexible modeling of cross-graph heterogeneity than is possible with conventional baseline (e.g., Bernoulli or $U|man$ distributions). We illustrate the utility of these baseline mixture models with application to problems of multiple-network ERGMs, network evolution, and efficient network inference. Our results underscore the potential ubiquity of network processes with nontrivial mixture behavior in natural settings, and raise some potentially disturbing questions regarding the adequacy of current network data collection practices.

Motivation & Objective

• Address the limitation of standard network models (e.g., Bernoulli or uniform graphs) in capturing cross-graph heterogeneity in network density or dyad frequencies.
• Develop analytically and computationally tractable network models that allow for flexible representation of variability across network realizations.
• Provide robust, weakly informative priors for network inference that perform well even when prior knowledge about network density is uncertain.
• Highlight the insufficiency of single cross-sectional network data for detecting mixture behavior in real-world network processes.
• Demonstrate that mixture priors can reduce data requirements for reliable inference, making them practical for typical empirical settings.

Proposed method

• Formalize network models as continuous mixtures of baseline distributions: beta-Bernoulli graphs for edge probabilities and Dirichlet-categorical graphs for dyad frequencies.
• Model network density as a random variable drawn from a beta distribution, leading to a beta-Bernoulli mixture that allows for flexible, heavy-tailed edge probability distributions.
• Model dyad configurations as drawn from a Dirichlet distribution, enabling the representation of reciprocity and other dyadic dependencies in a tractable way.
• Use hierarchical modeling to represent network data as arising from a mixture process where the baseline parameters (e.g., expected density) are themselves random variables.
• Apply these mixture models as priors in network inference, particularly in settings with limited observations per dyad.
• Leverage the conjugacy of the beta and Dirichlet distributions to enable efficient posterior computation and analytical tractability.

Experimental results

Research questions

• RQ1 Can continuous mixtures of baseline graph distributions (e.g., Bernoulli or uniform) better capture cross-graph heterogeneity than fixed-parameter models?
• RQ2 How do beta-Bernoulli and Dirichlet-categorical mixture models perform as priors in network inference when prior knowledge about network density is weak or uncertain?
• RQ3 What is the minimum number of observations per dyad required for mixture priors to achieve reliable inference compared to standard Bernoulli priors?
• RQ4 To what extent do current single-cross-sectional network data collection practices fail to detect mixture behavior arising from dynamic network processes?
• RQ5 Can mixture priors exploit dyadic dependence (e.g., reciprocity) to improve inference efficiency and robustness?

Key findings

• Beta-Bernoulli and Dirichlet-categorical mixture models significantly improve robustness and efficiency in network inference compared to standard Bernoulli priors, especially when prior knowledge about network density is limited.
• The Dirichlet-categorical prior outperforms Bernoulli priors by pooling information across dyads, leveraging reciprocity to improve confidence in edge state predictions.
• With as few as 2–3 observations per dyad, the Dirichlet-categorical prior achieves strong performance, suggesting that mixture priors can reduce data requirements substantially.
• Even with only 5 observations per dyad, mixture models achieve good inference performance under realistic error rates, far fewer than the 20–35 observations typical in full cross-sectional survey (CSS) designs.
• The results indicate that most extant social network data—collected as single cross-sectional snapshots—is insufficient to detect underlying mixture behavior in network processes.
• Baseline mixture models serve as superior default priors in network inference due to their flexibility, robustness, and ability to handle uncertainty in network density.