  This article focuses on the problem of studying shared- and
individual-specific structure in replicated networks or graph-valued data. In
particular, the observed data consist of $n$ graphs, $G_i, i=1,\ldots,n$, with
each graph consisting of a collection of edges between $V$ nodes. In brain
connectomics, the graph for an individual corresponds to a set of
interconnections among brain regions. Such data can be organized as a $V \times
V$ binary adjacency matrix $A_i$ for each $i$, with ones indicating an edge
between a pair of nodes and zeros indicating no edge. When nodes have a shared
meaning across replicates $i=1,\ldots,n$, it becomes of substantial interest to
study similarities and differences in the adjacency matrices. To address this
problem, we propose a method to estimate a common structure and low-dimensional
individual-specific deviations from replicated networks. The proposed Multiple
GRAph Factorization (M-GRAF) model relies on a logistic regression mapping
combined with a hierarchical eigenvalue decomposition. We develop an efficient
algorithm for estimation and study basic properties of our approach. Simulation
studies show excellent operating characteristics and we apply the method to
human brain connectomics data.

Computation