Generating random partial correlation matrices with an application to redundant variables and bridge centrality

The Gaussian graphical model (GGM) estimates partial correlations among variables and is a popular network model for psychological symptom data. Centrality indices are functions of partial correlations and are commonly used to interpret the importance of variables in fitted networks. For instance, bridge centrality quantifies the extent to which a variable (or symptom) has connections with variables in other communities (or symptom clusters) and may aid in understanding comorbidity of mental disorders. While critiques have emerged concerning the stability of centrality indices under changes to network composition, methods to simulate and study networks under some conditions of interest are underdeveloped. For example, extant approaches do not easily accommodate custom range restrictions on individual matrix elements or constraints on the pattern of partial correlations across the matrix, which limits researcher control over the data space and makes bridge centrality difficult to study. We adapted a Markov Chain Monte Carlo (MCMC) approach to generate random partial correlation matrices from a space constrained based on user-imposed specifications. We examined our MCMC method with two clusters of variables in which one variable had high bridge centrality but was highly correlated with another variable. We fit GGMs to the random partial correlation matrices and evaluated changes in bridge strength under removal of one of these variables. MCMC had similar coverage of the intended data space compared to a uniform sampling approach, but was faster to generate acceptable matrices. Bridge strength showed good stability and generally changed in an interpretable direction when one variable was removed.