Information flow in interaction networks II: channels, path lengths and potentials

In our previous publication, a framework for information flow in interaction networks based on random walks with damping was formulated with two fundamental modes: emitting and absorbing. While many other network analysis methods based on random walks or equivalent notions have been developed before and after our earlier work, one can show that they can all be mapped to one of the two modes. In addition to these two fundamental modes, a major strength of our earlier formalism was its accommodation of context-specific directed information flow that yielded plausible and meaningful biological interpretation of protein functions and pathways. However, the directed flow from origins to destinations was induced via a potential function that was heuristic. Here, with a theoretically sound approach called the channel mode, we extend our earlier work for directed information flow. This is achieved by constructing a potential function facilitating a purely probabilistic interpretation of the channel mode. For each network node, the channel mode combines the solutions of emitting and absorbing modes in the same context, producing what we call a channel tensor. The entries of the channel tensor at each node can be interpreted as the amount of flow passing through that node from an origin to a destination. Similarly to our earlier model, the channel mode encompasses damping as a free parameter that controls the locality of information flow. Through examples involving the yeast pheromone response pathway, we illustrate the versatility and stability of our new framework.Comment: Minor changes from v3. 30 pages, 7 figures. Plain LaTeX format. This version contains some additional material compared to the journal submission: two figures, one appendix and a few paragraph

On graph kernels: Hardness results and efficient alternatives

As most ‘real-world’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances that are represented by graphs. So far, only very specific graphs such as trees and strings have been considered. This paper investigates kernels on labeled directed graphs with general structure. It is shown that computing a strictly positive definite graph kernel is at least as hard as solving the graph isomorphism problem. It is also shown that computing an inner product in a feature space indexed by all possible graphs, where each feature counts the number of subgraphs isomorphic to that graph, is NP-hard. On the other hand, inner products in an alternative feature space, based on walks in the graph, can be computed in polynomial time. Such kernels are defined in this paper