Inference for sender-receiver alignment in directed networks

In a directed network, does understanding a node's sending behaviour give us any information about its receiving behaviour? To address this question, we propose the effective shared dimension, a real-valued count of how many directions the two roles share, alongside a degree-free counterpart which removes the effect of a node's overall activity level. We define both for general latent position models, and demonstrate that under a directed random dot product graph model they are explicit functions of moments of the underlying node positions. For each quantity we give an estimator, for which we prove consistency and, provided the average node degree grows faster than $n^{1/2}$, asymptotic normality, before discussing a bias correction to potentially extend the latter to sparser regimes. Finally, we apply this methodology to estimate the effective shared dimension across 10 real directed networks spanning social, information, infrastructure and biological data, and ask whether the answer differs systematically between these domains.