Causal inference methods for treatment effect estimation usually assume independent units. However, this assumption is often questionable because units may interact, resulting in spillover effects between them. We develop augmented inverse probability weighting (AIPW) for estimation and inference of the expected average treatment effect (EATE) with observational data from a single (social) network with spillover effects. In contrast to overall effects such as the global average treatment effect (GATE), the EATE measures, in expectation and on average over all units, how the outcome of a unit is causally affected by its own treatment, marginalizing over the spillover effects from other units. We develop cross-fitting theory with plugin machine learning to obtain a semiparametric treatment effect estimator that converges at the parametric rate and asymptotically follows a Gaussian distribution. The asymptotics are developed using the dependency graph rather than the network graph, which makes explicit that we allow for spillover effects beyond immediate neighbors in the network. We apply our AIPW method to the Swiss StudentLife Study data to investigate the effect of hours spent studying on exam performance accounting for the students' social network.