INTRODUCTION
aphasia
REGRESSION, APPENDIX
The study of resting‐state brain connectivity via functional magnetic resonance imaging (fMRI) involves the investigation of correlations between cortical seeds, regions, or voxels (henceforth referred to as foci). Friston, in particular, defined functional connectivity as the correlations, over time, between spatially distinct brain regions (Friston, To illustrate the idea of label invariance, consider a scenario where one reduces the connectivity measures to subject‐specific binary graphs (by thresholding). If the effect of the graphs on the outcomes is invariant to the nodes (foci) corresponding to the edges, then clearly it is sufficient to know the number of edges that are present for each subject's graph, since given that information one can create the set of equivalent graphs under node invariance. This is equivalent to saying the relationship between the outcome and connectivity graph, is solely dependent on the estimated probability distribution for the edges under an assumed independent and identically distributed edge distribution, since that distribution only depends on the total number of edges. (This is the Erdős‐Rényi random graph model.) Our approach formally builds on this idea. But we further consider a random weighted graph model rather than thresholding to obtain binary edges, and propose a specific functional linear model for the relationship between outcomes and the connectivity density.We demonstrate the benefits of using the distribution of resting state correlations as covariates using functional data analysis tools. We suggest the use of the quantile density, the density of connections evaluated at evenly spaced quantiles of the connections, as this improves performance. Regardless of these choices, utilizing connectivity density regression has several benefits. A primary one is the relaxation of the consistent localization assumption across subjects. In the Appendix Hyperalignment (Haxby et al., There are existing studies that utilize the distribution of resting state correlations. For example, Petersen and Muüller (This study is motivated by a resting‐state fMRI study of primary progressive aphasia (PPA) patients, where it is feasible to want to relax the geometric localization assumption. In the study, the patients were randomly assigned into two treatment groups, (a) transcranial direct‐current stimulation (tDCS; Nitsche et al., The manuscript is organized as follows. In Section 