  In this paper, we introduce a discrete time-finite state model for pedestrian
flow on a graph in the spirit of the Hughes dynamic continuum model. The
pedestrians, represented by a density function, move on the graph choosing a
route to minimize the instantaneous travel cost to the destination. The density
is governed by a conservation law while the minimization principle is described
by a graph eikonal equation. We show that the model is well posed and we
implement some numerical examples to demonstrate the validity of the proposed
model.
