Long-Range Navigation on Complex Networks using Levy Random Walks

We introduce a strategy of navigation in undirected networks, including regular, random, and complex networks, that is inspired by L'evy random walks, generalizing previous navigation rules. We obtained exact expressions for the stationary probability distribution, the occupation probability, the mean first passage time, and the average time to reach a node on the network. We found that the long-range navigation using the L'evy random walk strategy, compared with the normal random walk strategy, is more efficient at reducing the time to cover the network. The dynamical effect of using the L'evy walk strategy is to transform a large-world network into a small world. Our exact results provide a general framework that connects two important fields: L'evy navigation strategies and dynamics on complex networks.

💡 Research Summary

The paper introduces a novel navigation strategy for undirected networks—regular lattices, Erdős‑Rényi random graphs, and scale‑free Barabási‑Albert networks—by incorporating Lévy‑type long‑range jumps into the random walk dynamics. In the classical simple random walk (SRW), a walker moves only to immediate neighbors, which leads to inefficient exploration on large‑world networks where the average shortest‑path length grows rapidly with system size. To overcome this limitation, the authors define a transition probability that decays as a power of the topological distance d between nodes:

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