Independent decisions collectively producing a long information dissemination path with a foreseen lower-bounded length in a network

Our research problems can be understood with the following metaphor: In Facebook or Twitter, suppose Mike decides to send a message to a friend Jack, and Jack next decides to pass the message to one of his own friends Mary, and the process continues until the current message holder could not find a friend who is not in the relaying path. How to make the message live longer in the network with each individual's local decision? Can Mike foresee the length of the longest paths starting with himself in the network by only collecting information of native nature? In contrast to similar network problems with respect to short paths, e.g., for explaining the famous Milgram's small world experiment, no nontrivial solutions have been proposed for the problems. The two research problems are not completely the same and notably our approach yields solutions to both. We discover node-specific numeric values which can be computed by only communicating identity-free degree derivatives to network neighbors and for an arbitrary network node $v$ there exists a function determining a lower bound for the length of the longest paths starting with it based on its numeric values. Moreover, in the navigation process initiated at $v$, inspecting the numeric values of their neighbors, the involved nodes can independently make their decisions eventually guaranteeing a path of length longer than the determined lower bound at $v$. Numerical analyses demonstrate plausible performance of our approach of inferring certain global properties from local information in complex networks.