  Network data have appeared frequently in recent research. For example, in
comparing the effects of different types of treatment, network models have been
proposed to improve the quality of estimation and hypothesis testing. In this
paper, we focus on efficiently estimating the average treatment effect using an
adaptive randomization procedure in networks. We work on models of causal
frameworks, for which the treatment outcome of a subject is affected by its own
covariate as well as those of its neighbors. Moreover, we consider the case in
which, when we assign treatments to the current subject, only the subnetwork of
existing subjects is revealed. New randomized procedures are proposed to
minimize the mean squared error of the estimated differences between treatment
effects. In network data, it is usually difficult to obtain theoretical
properties because the numbers of nodes and connections increase
simultaneously. Under mild assumptions, our proposed procedure is closely
related to a time-varying inhomogeneous Markov chain. We then use Lyapunov
functions to derive the theoretical properties of the proposed procedures. The
advantages of the proposed procedures are also demonstrated by extensive
simulations and experiments on real network data.
