  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.
