Multilevel Aggregation Methods for Small-World Graphs with Application to Random-Walk Ranking

We describe multilevel aggregation in the specific context of using Markov chains to rank the nodes of graphs. More generally, aggregation is a graph coarsening technique that has a wide range of possible uses regarding information retrieval applications. Aggregation successfully generates efficient multilevel methods for solving nonsingular linear systems and various eigenproblems from discretized partial differential equations, which tend to involve mesh-like graphs. Our primary goal is to extend the applicability of aggregation to similar problems on small-world graphs, with a secondary goal of developing these methods for eventual applicability towards many other tasks such as using the information in the hierarchies for node clustering or pattern recognition. The nature of small-world graphs makes it difficult for many coarsening approaches to obtain useful hierarchies that have complexity on the order of the number of edges in the original graph while retaining the relevant properties of the original graph. Here, for a set of synthetic graphs with the small-world property, we show how multilevel hierarchies formed with non-overlapping strength-based aggregation have optimal or near optimal complexity. We also provide an example of how these hierarchies are employed to accelerate convergence of methods that calculate the stationary probability vector of large, sparse, irreducible, slowly-mixing Markov chains on such small-world graphs. The stationary probability vector of a Markov chain allows one to rank the nodes in a graph based on the likelihood that a long random walk visits each node. These ranking approaches have a wide range of applications including information retrieval and web ranking, performance modeling of computer and communication systems, analysis of social networks, dependability and security analysis, and analysis of biological systems.