The mutually-exciting structure of the Hawkes process makes it particularly suitable for modelling real-world networks in neuroscience, high-frequency finance, genomics and social network analysis. There is now a growing interest in developing adaptive (or online) algorithms suitable for streaming data and also to deal with time-variant parameters in offline data. Adaptive estimation for the Hawkes process is challenging due to non-negativity constraints on the parameters. In this paper, we overcome this by modelling the vector log-stochastic intensity and then develop a fixed gain adaptive distributed estimator based on the point process instantaneous likelihood. We apply the algorithm to some genomic data and find evidence of time-varying parameters. This seems to be the first example of its kind.