  This paper studies the properties of linear regression on centrality measures
when network data is sparse -- that is, when there are many more agents than
links per agent -- and when they are measured with error. We make three
contributions in this setting: (1) We show that OLS estimators can become
inconsistent under sparsity and characterize the threshold at which this
occurs, with and without measurement error. This threshold depends on the
centrality measure used. Specifically, regression on eigenvector is less robust
to sparsity than on degree and diffusion. (2) We develop distributional theory
for OLS estimators under measurement error and sparsity, finding that OLS
estimators are subject to asymptotic bias even when they are consistent.
Moreover, bias can be large relative to their variances, so that bias
correction is necessary for inference. (3) We propose novel bias correction and
inference methods for OLS with sparse noisy networks. Simulation evidence
suggests that our theory and methods perform well, particularly in settings
where the usual OLS estimators and heteroskedasticity-consistent/robust t-tests
are deficient. Finally, we demonstrate the utility of our results in an
application inspired by De Weerdt and Deacon (2006), in which we consider
consumption smoothing and social insurance in Nyakatoke, Tanzania.
