  This article proposes a Bayesian approach to regression with a continuous
scalar response and an undirected network predictor. Undirected network
predictors are often expressed in terms of symmetric adjacency matrices, with
rows and columns of the matrix representing the nodes, and zero entries
signifying no association between two corresponding nodes. Network predictor
matrices are typically vectorized prior to any analysis, thus failing to
account for the important structural information in the network. This results
in poor inferential and predictive performance in presence of small sample
sizes. We propose a novel class of network shrinkage priors for the coefficient
corresponding to the undirected network predictor. The proposed framework is
devised to detect both nodes and edges in the network predictive of the
response. Our framework is implemented using an efficient Markov Chain Monte
Carlo algorithm. Empirical results in simulation studies illustrate strikingly
superior inferential and predictive gains of the proposed framework in
comparison with the ordinary high dimensional Bayesian shrinkage priors and
penalized optimization schemes. We apply our method to a brain connectome
dataset that contains information on brain networks along with a measure of
creativity for multiple individuals. Here, interest lies in building a
regression model of the creativity measure on the network predictor to identify
important regions and connections in the brain strongly associated with
creativity. To the best of our knowledge, our approach is the first principled
Bayesian method that is able to detect scientifically interpretable regions and
connections in the brain actively impacting the continuous response
(creativity) in the presence of a small sample size.
