The collective attention on online items such as web pages, search terms, and videos reflects trends that are of social, cultural, and economic interest. Moreover, attention trends of different items exhibit mutual influence via mechanisms such as hyperlinks or recommendations. Many visualisation tools exist for time series, network evolution, or network influence; however, few systems connect all three. In this work, we present AttentionFlow, a new system to visualise networks of time series and the dynamic influence they have on one another. Centred around an ego node, our system simultaneously presents the time series on each node using two visual encodings: a tree ring for an overview and a line chart for details. AttentionFlow supports interactions such as overlaying time series of influence, and filtering neighbours by time or flux. We demonstrate AttentionFlow using two real-world datasets, VevoMusic and WikiTraffic. We show that attention spikes in songs can be explained by external events such as major awards, or changes in the network such as the release of a new song. Separate case studies also demonstrate how an artist's influence changes over their career, and that correlated Wikipedia traffic is driven by cultural interests. More broadly, AttentionFlow can be generalised to visualise networks of time series on physical infrastructures such as road networks, or natural phenomena such as weather and geological measurements.