Diffusion Kernels

Graphs are some one of the simplest type of objects in Mathematics. In Chapter Chapter ?? we saw how to construct kernels between graphs, that is, when the individual examples x ∈ X are graphs. In this chapter we consider the case when the input space X is itself a graph and the examples are vertices in this graph. Such a case may arise naturally in diverse contexts. We may be faced with a network, trying to extrapolate known values of some quantity at specific nodes to other nodes. An illustrative example might be the graph of interactions between the proteins of an organism, which can be built from recent high-throughput technologies. Let's say we are trying to learn the localization of proteins in the cell, or their functions. In the absence of other knowledge, a reasonable starting point is to assume that proteins that can interact are likely to have similar localization or functions. Other examples of naturally occuring networks include metabolic or signalling pathways, and also social networks, the World Wide Web, or citation networks. In other cases we might be faced with something much more intricate that is not itself a network, but can conveniently be modelled as one. Assume we are interested in prediciting the physical properties of organic molecules. Clearly, the set of all known organic molecules is very large and it is next to impossible to impose a global metric on it or sensibly fit it into a vector space. On the other hand, it is not so