Dynamic degree-corrected blockmodels for social networks: A nonparametric approach

Abstract

A nonparametric approach to the modeling of social networks using degreecorrected stochastic blockmodels is proposed. The model for static network consists of a stochastic blockmodel using a probit regression formulation and popularity parameters are incorporated to account for degree heterogeneity. Dirichlet processes are used to detect community structure as well as induce clustering in the popularity parameters. This approach is flexible yet parsimonious as it allows the appropriate number of communities and popularity clusters to be determined automatically by the data. We further discuss some ways of extending the static model to dynamic networks. We consider a Bayesian approach and derive Gibbs samplers for posterior inference. The models are illustrated using several real-world benchmark social networks.

Bayesian Learning of Graph Substructures

“…We repeat the simulation for n = 10 4 , 10 3 , 10 2 , 10 while keeping Ω the same and present the estimated community structure in Figure 2. For n = 10 4 , the results are very close to those in Tan and De Iorio (2019) where the underlying network is known, with the two main blocks corresponding to the karate instructor Mr Hi and the club's president John A. The increased uncertainty in the estimation of G for smaller values of n obviously affects inference on the block structure, with too little information present in the data with only n = 10 observations to recover the two main blocks.…”

“…In general, these approaches require also specification of prior edge inclusion probabilities jointly with the block structure prior. For instance, Kemp et al (2006), Geng et al (2018 and Legramanti et al (2022b) place Beta distributions on the edge probabilities, and Reyes and Rodríguez (2016) and Jiang and Tokdar (2021) add structure by using different priors for within-and between-block edge probabilities, while Tan and De Iorio (2019) use a DP to build a joint prior on the partition of nodes and edge probabilities. Additionally, they extend the model to a degree-corrected blockmodel, i.e.…”

“…Our starting point is the Bayesian nonparametric degree-corrected stochastic blockmodel by Tan and De Iorio (2019) who propose a probit model for the edge inclusion probabilities. More specifically, Pr{(i, j) ∈ E} = Φ(μ ij ), independently over distinct pairs (i, j) for 1 ≤ i < j ≤ p, where Φ(•) is the cumulative distribution function of the standard Gaussian distribution N (0, 1).…”

“…We investigate the importance of uncertainty propagation when learning community structure in a graphical model. As true underlying graph G, we consider the karate club network (Zachary, 1977) which Tan and De Iorio (2019) analyse using the degreecorrected stochastic blockmodel of Section 3.1. The network's p = 34 nodes correspond to members of a karate club while its 78 edges signify friendships between members.…”

“…Finally, we fit the model from Section 3.1 using 6000 MCMC iterations, discarding the first 1000 as burn-in. In this case, we set a ν = b ν = a α = b α = 5 as in Tan and De Iorio (2019) for a fair comparison.…”