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It is a pleasure to congratulate Ni et al. (Stat Methods Appl 490:1–32, 2021) on the recent advances in Bayesian graphical models reviewed in Ni et al. (Stat Methods Appl 490:1–32, 2021). The authors have given considerable thought to the construction and estimation of Bayesian graphical models that capture salient features of biological networks. My discussion focuses on computational challenges and opportunities along with priors, pointing out limitations of the Markov random field priors reviewed in Ni et al. (Stat Methods Appl 490:1–32, 2021) and exploring possible generalizations that capture additional features of conditional independence graphs, such as hub structure and clustering. I conclude with a short discussion of the intersection of graphical models and random graph models. 

We consider incomplete observations of stochastic processes governing the spread of infectious diseases through finite populations by way of contact. We propose a flexible semiparametric modeling framework with at least three advantages. First, it enables researchers to study the structure of a population contact network and its impact on the spread of infectious diseases. Second, it can accommodate short- and long-tailed degree distributions and detect potential superspreaders, who represent an important public health concern. Third, it addresses the important issue of incomplete data. Starting from first principles, we show when the incomplete-data generating process is ignorable for the purpose of Bayesian inference for the parameters of the population model. We demonstrate the semiparametric modeling framework by simulations and an application to the partially observed MERS epidemic in South Korea in 2015. We conclude with an extended discussion of open questions and directions for future research. 

A widely used approach to modeling discrete‐time network data assumes that discrete‐time network data are generated by an unobserved continuous‐time Markov process. While such models can capture a wide range of network phenomena and are popular in social network analysis, the models are based on the homogeneity assumption that all nodes share the same parameters. We remove the homogeneity assumption by allowing nodes to belong to unobserved subsets of nodes, called blocks, and assuming that nodes in the same block have the same parameters, whereas nodes in distinct blocks have distinct parameters. The resulting models capture unobserved heterogeneity across nodes and admit model‐based clustering of nodes based on network properties chosen by researchers. We develop Bayesian data‐augmentation methods and apply them to discrete‐time observations of an ownership network of non‐financial companies in Slovenia in its critical transition from a socialist economy to a market economy. We detect a small subset of shadow‐financial companies that outpaces others in terms of the rate of change and the desire to accumulate stocks of other companies.

 

Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple example of a random graph with additional structure is a random graph with neighborhoods and local dependence within neighborhoods. We develop the first concentration and consistency results for maximum likelihood and M-estimators of a wide range of canonical and curved exponentialfamily models of random graphs with local dependence. All results are nonasymptotic and applicable to random graphs with finite populations of nodes, although asymptotic consistency results can be obtained as well. In addition, we show that additional structure can facilitate subgraph-to-graph estimation, and present concentration results for subgraph-to-graph estimators. As an application, we consider popular curved exponential-family models of random graphs, with local dependence induced by transitivity and parameter vectors whose dimensions depend on the number of nodes. 

Multilevel network data provide two important benefits for ERG modeling. First, they facilitate estimation of the decay parameters in geometrically weighted terms for degree and triad distributions. Estimating decay parameters from a single network is challenging, so in practice they are typically fixed rather than estimated. Multilevel network data overcome that challenge by leveraging replication. Second, such data make it possible to assess out-ofsample performance using traditional cross-validation techniques. We demonstrate these benefits by using a multilevel network sample of classroom networks from Poland. We show that estimating the decay parameters improves in-sample performance of the model and that the out-of-sample performance of our best model is strong, suggesting that our findings can be generalized to the population of interest. 

Social network data are complex and dependent data. At the macro-level, social networks often exhibit clustering in the sense that social networks consist of communities; and at the micro-level, social networks often exhibit complex network features such as transitivity within communities. Modeling real-world social networks requires modeling both the macro- and micro-level, but many existing models focus on one of them while neglecting the other. In recent work, [28] introduced a class of Exponential Random Graph Models (ERGMs) capturing community structure as well as microlevel features within communities. While attractive, existing approaches to estimating ERGMs with community structure are not scalable. We propose here a scalable two-stage strategy to estimate an important class of ERGMs with community structure, which induces transitivity within communities. At the first stage, we use an approximate model, called working model, to estimate the community structure. At the second stage, we use ERGMs with geometrically weighted dyadwise and edgewise shared partner terms to capture refined forms of transitivity within communities. We use simulations to demonstrate the performance of the two-stage strategy in terms of the estimated community structure. In addition, we show that the estimated ERGMs with geometrically weighted dyadwise and edgewise shared partner terms within communities outperform the working model in terms of goodness-of-fit. Last, but not least, we present an application to high-resolution human contact network data.