  In this paper, we investigate the Gaussian graphical model inference problem
in a novel setting that we call erose measurements, referring to irregularly
measured or observed data. For graphs, this results in different node pairs
having vastly different sample sizes which frequently arises in data
integration, genomics, neuroscience, and sensor networks. Existing works
characterize the graph selection performance using the minimum pairwise sample
size, which provides little insights for erosely measured data, and no existing
inference method is applicable. We aim to fill in this gap by proposing the
first inference method that characterizes the different uncertainty levels over
the graph caused by the erose measurements, named GI-JOE (Graph Inference when
Joint Observations are Erose). Specifically, we develop an edge-wise inference
method and an affiliated FDR control procedure, where the variance of each edge
depends on the sample sizes associated with corresponding neighbors. We prove
statistical validity under erose measurements, thanks to careful localized
edge-wise analysis and disentangling the dependencies across the graph.
Finally, through simulation studies and a real neuroscience data example, we
demonstrate the advantages of our inference methods for graph selection from
erosely measured data.
