The diffusion kernel is a general method for computing pairwise distances among all nodes in a graph, based upon the sum of weighted paths between each pair of nodes.  This technique has been used successfully, in conjunction with kernel-based learning methods, to draw inferences from several types of biological networks.  We show that computing the diffusion kernel is equivalent to maximizing the von Neumann entropy, subject to a global constraint on the sum of the Euclidean distances between nodes.  This global constraint allows for high variance in the pairwise distances.  Accordingly, we propose an alternative, locally constrained diffusion kernel, and we demonstrate that the resulting kernel allows for more accurate support vector machine prediction of protein functional classifications from metabolic and protein-protein interaction networks.