  Network data are increasingly available in various research fields,
motivating statistical analysis for populations of networks where a network as
a whole is viewed as a data point. Due to the non-Euclidean nature of networks,
basic statistical tools available for scalar and vector data are no longer
applicable when one aims to relate networks as outcomes to Euclidean
covariates, while the study of how a network changes in dependence on
covariates is often of paramount interest. This motivates to extend the notion
of regression to the case of responses that are network data. Here we propose
to adopt conditional Fr\'{e}chet means implemented with both global least
squares regression and local weighted least squares smoothing, extending the
Fr\'{e}chet regression concept to networks that are quantified by their graph
Laplacians. The challenge is to characterize the space of graph Laplacians so
as to justify the application of Fr\'{e}chet regression. This characterization
then leads to asymptotic rates of convergence for the corresponding
M-estimators by applying empirical process methods. We demonstrate the
usefulness and good practical performance of the proposed framework with
simulations and with network data arising from resting-state fMRI in
neuroimaging, as well as New York taxi records.
