Reducing Search Lengths with Locally Precomputed Partial Random Walks

Random walks can be used to search complex networks for a desired resource. To reduce the number of hops necessary to find resources, we propose a search mechanism based on building random walks connecting together partial walks that have been precomputed at each network node in an initial stage. The resources found in each partial walk are registered in its associated Bloom filter. Searches can then jump over partial nodes in which the resource is not located, significantly reducing search length. However, additional unnecessary hops come from false positives at the Bloom filters. Two variations of the mechanism just described have been considered, differing in the type of partial walks computed in the initial stage: simple random walks or self-avoiding random walks. Analytical models have been developed to predict the expected search length of these mechanisms. When partial walks are random walks, the model also provides expressions for the optimal size of the partial walks and the corresponding optimal (shortest) expected search length. We have found that the optimal search length is proportional to the square root of the expected length of searches based on simple random walks, achieving a significant improvement. Further reductions are obtained when partial walks are self-avoiding random walks. Simulation experiments are used to validate these predictions and to assess the impact of the number of partial walks precomputed in each node. We have found that with just two partial walks per node the results are similar to those obtained for larger values, which is a significant result regarding the practical implementation of the search mechanism.