Network theory provides a principled abstraction of the human brain: reducing a complex system into a simpler representation from which to investigate brain organisation. Recent advancement in the neuroimaging field are towards representing brain connectivity as a dynamic process in order to gain a deeper understanding of the interplay between functional modules for efficient information transport. In this work, we employ heat kernels to model the process of energy diffusion in functional networks. We extract node-based, multi-scale features which describe the propagation of heat over 'time' which not only inform the importance of a node in the graph, but also incorporate local and global information of the underlying geometry of the network. As a proof-of-concept, we test the efficacy of two heat kernel features for discriminating between motor and working memory functional networks from the Human Connectome Project. For comparison, we also classified task networks using traditional network metrics which similarly provide rankings of node importance. In addition, a variant of the Smooth Incremental Graphical Lasso Estimation algorithm was used to estimate non-sparse, precision matrices to account for non-stationarity in the time series. We illustrate differences in heat kernel features between tasks, and also between regions of the brain. Using a random forest classifier, we showed heat kernel metrics to capture intrinsic properties of functional networks that serve well as features for task classification
