Principal component analysis (PCA) serves as a fundamental statistical technique for such analyses.  However, performing PCA on high-dimensional connectivity matrices yields complicated "eigenconnectivity" patterns, for which systematic interpretation is a challenging issue.  Here, we overcome this issue with a novel constrained PCA method for connectivity matrices by extending the idea of the previously proposed orthogonal connectivity factorization method.  We develop an optimization algorithm to solve the constrained PCA problem and validate our method in simulation studies and with a resting-state functional connectivity MRI dataset of 986 subjects. 
PCA
Our new method, modular connectivity factorization (MCF), explicitly introduces the modularity of brain networks as a parametric constraint on eigenconnectivity matrices.  In particular, MCF analyzes the variability in both intra- and inter-module connectivities, simultaneously finding network modules in a principled, data-driven manner.  The results show that the proposed MCF method successfully reveals the underlying modular eigenconnectivity patterns in more general situations and is a promising alternative to existing methods.
MCF
Here, we overcome this issue with a novel constrained PCA method for connectivity matrices by extending the idea of the previously proposed orthogonal connectivity factorization method. 
orthogonal connectivity factorization