Suppose there is a message generated at a node $v$ in a network and $v$ decides to pass the message to one of the neighbors $u$, and $u$ next decides to pass the message to one of its own neighbors, and so on. How to relay the message as far as possible with local decisions? To the best of our knowledge no general solution other than randomly picking available adjacent node exists. Here we report some progress. Our first contribution is a new framework called tp-separate chain decomposition for studying network structures. Each tp-separate chain induces a ranking of nodes. We then prove that the ranks can be locally and distributively computed via searching some stable states of certain dynamical systems on the network and can be used to search long paths of a guaranteed length containing any given node. Numerical analyses on a number of typical real-world networks demonstrate the effectiveness of the approach.
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