  We introduce the concept of a Markov influence system (MIS) and analyze its
dynamics. An MIS models a random walk in a graph whose edges and transition
probabilities change endogenously as a function of the current distribution.
This article consists of two independent parts: in the first one, we generalize
the standard classification of Markov chain states to the time-varying case by
showing how to "parse" graph sequences; in the second part, we use this
framework to carry out the bifurcation analysis of a few important MIS
families. We show that, in general, these systems can be chaotic but that
irreducible MIS are almost always asymptotically periodic. We give an example
of "hyper-torpid" mixing, where a stationary distribution is reached in
super-exponential time, a timescale beyond the reach of any Markov chain.
