On Graph Isomorphism and the PageRank Algorithm

Graphs express relationships among objects, such as the radio connectivity among nodes in unmanned vehicle swarms. Some applications may rank a swarms nodes by their relative importance, for example, using the PageRank algorithm applied in certain search engines to order query responses. The PageRank values of the nodes correspond to a unique eigenvector that can be computed using the power method, an iterative technique based on matrix multiplication. The first result is a practical lower bound on the PageRank algorithms execution time that is derived by applying assumptions to the PageRank perturbations scaling value and the PageRank vectors required numerical precision. The second result establishes nodes contained in the same block of the graphs coarsest equitable partition must have equal PageRank values. The third result, the AverageRank algorithm, ensures such nodes are assigned equal PageRank values. The fourth result, the ProductRank algorithm, reduces the time needed to find the PageRank vector by eliminating certain dot products in the power method if the graphs coarsest equitable partition contains blocks composed of multiple vertices. The fifth result, the QuotientRank algorithm, uses a quotient matrix induced by the coarsest equitable partition to further reduce the time needed to compute a swarms PageRank vector.