Behavioural differences among social groups can arise from differing ecological conditions, genetic predispositions and/or social learning. In the past, social learning has typically been inferred as responsible for the spread of behaviour by the exclusion of ecological and genetic factors. This 'method of exclusion' was used to infer that 'sponging', a foraging behaviour involving tool use in the bottlenose dolphin (Tursiops aduncus) population in Shark Bay, Western Australia, was socially transmitted. However, previous studies were limited in that they never fully accounted for alternative factors, and that social learning, ecology and genetics are not mutually exclusive in causing behavioural variation. Here, we quantified the importance of social learning on the diffusion of sponging, for the first time explicitly accounting for ecological and genetic factors, using a multi-network version of 'network-based diffusion analysis'. Our results provide compelling support for previous findings that sponging is vertically socially transmitted from mother to (primarily female) offspring. This research illustrates the utility of social network analysis in elucidating the explanatory mechanisms behind the transmission of behaviour in wild animal populations.