This paper presents a dynamic network autoregressive conditional heteroscedasticity (ARCH) model suitable for high-dimensional cases where multivariate ARCH models are typically no longer applicable. We adopt the theoretical foundations from spatiotemporal statistics and transfer the dynamic ARCH model processes to networks. The model integrates temporally lagged volatility and information from adjacent nodes, which may instantaneously spill across the entire network. The model is used to forecast volatility in the US stock market, and the edges are determined based on various distance and correlation measures between the time series. The performance of alternative network definitions is compared with independent univariate log-ARCH models in terms of out-of-sample prediction accuracy. The results indicate that more accurate forecasts are obtained with network-based models and that accuracy can be improved by combining the forecasts of different network definitions. We emphasise the significance for practitioners to integrate network structure information when developing volatility forecasts.