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The concept of intransitiveness for games, which is the condition for whichthere is no first-player winning strategy can arise surprisingly, as happens inthe Penney game, an extension of the heads or tails. Since a game can beconverted into a random walk on a graph, i.e., a Markov process, we extend theintransitiveness concept to such systems. The end of the game generallyconsists in the appearance of a pre-defined pattern. In the language of randomwalk this corresponds to an absorbing trap, since once that the game hasreached this condition the game comes to an end. Therefore, theintransitiveness of the game can be mapped into a problem of competition amongtraps. We analyse in details random walkers on several kind of networks (rings,scale-free, hierarchical and city-inspired) with several variations: traps canbe partially absorbing, the walker can be biased and the initial distributioncan be arbitrary. We found that the transitivity concept can be quite usefulfor characterizing the combined properties of a graph and that of the walkers.

Intransitiveness in the Penney Game and in Random Walks on rings, networks, communities and cities