We propose a new model for networks of time series that influence each other. Graph structures among time series is found in diverse domains, such as web traffic influenced by hyperlinks, product sales influenced by recommendation, or urban transport volume influenced by the road network and weather. There has been recent progress in modeling graphs and time series, respectively, but an expressive and scalable approach for a network of series does not yet exist. We introduce Radflow, a novel model that embodies three main ideas: the recurrent structure of LSTM to obtain time- dependent node embeddings, aggregation of the flow of influence from other nodes with multi-head attention, and multi-layer decomposition of time series. Radflow naturally takes into account dynamic networks where nodes and edges appear over time, and it can be used for prediction and data imputation tasks. On four real-world datasets ranging from a few hundred to a few hundred thousand nodes, we observe Radflow variants being the best performing model across all tasks. We also report that the recurrent component in Radflow consistently outperforms N-BEATS, the state-of-the-art time series model. We show that Radflow can learn different trends and seasonal patterns, that it is robust to missing nodes and edges, and that correlated temporal patterns among network neighbors reflect influence strength. We curate WikiTraffic, the largest dynamic network of time series with 360K nodes and 22M time-dependent links spanning five years—this dataset provides an open benchmark for developing models in this area, and prototyping applications for problems such as estimating web resources and optimizing collaborative infrastructures. More broadly, Radflow can be used to improve the forecasts in correlated time series networks such as the stock market, or impute missing measurements of natural phenomena such as geographically dispersed networks of waterbodies.