  We present continuum models that describe the evolution of the position of a
random walker on a growing network using four different growth algorithms.
Three of these involve a random element, including one in which the motility
rate of the random walker controls the network topology. For motility rates in
which the position of the walker can be treated as quasi-stationary, we present
accurate approximations to replace pair probabilities that allow us to
numerically solve an otherwise intractable system of equations.
