  We find that the simple coupling of network growth to the position of a
random walker on the network generates a traveling wave in the probability
distribution of nodes visited by the walker. We argue that the entropy of this
probability distribution is bounded as the network size tends to infinity. This
means that the growth of a space coupled to a random walker situated in it
constrains its dynamics to a set of typical random walker trajectories, and
walker trajectories inside the growing space are compressible.
