Identifying signs of online extremism is one of the top priorities for counter-extremist agencies. Social media platforms have become prime locations for radicalisation content and behaviour, and therefore much research and practice nowadays are focused on detecting radicalisation material, and accounts that publish such material, on these platforms. However, there is currently a limited understanding of how people on social media platforms are influenced by such content and behaviour, and what are the dynamics of this influence. In this paper, we propose a computational approach for detecting and predicting the radicalisation influence that a user is subjected to. Our approach is grounded on the notion of ‘roots of radicalisation’ from social science theories. We use our approach to analyse and compare the radicalisation influence of 112 pro-ISIS and 112 “general” Twitter users. Our results show the effectiveness of our proposed algorithms in detecting and predicting radicalisation influence, obtaining up to 0.9 F-1 measure for detection and between 0.7 and 0.8 precision for prediction. We have also conducted an in-depth analysis of the social influence received by the 112 pro-ISIS accounts, and reported on the origin, frequency and topical diversity of this influence. While this is an initial attempt towards the effective combination of social and computational perspectives, more work is needed to bridge these disciplines, and to build on their strengths to target the problem of online radicalisation.