  We present a mean-field theory for the dynamics of driven flow with exclusion
in graphenelike structures, and numerically check its predictions. We treat
first a specific combination of bond transmissivity rates, where mean field
predicts, and numerics to a large extent confirms, that the sublattice
structure characteristic of honeycomb networks becomes irrelevant. Dynamics, in
the various regions of the phase diagram set by open boundary injection and
ejection rates, is then in general identical to that of one-dimensional
systems, although some discrepancies remain between mean-field theory and
numerical results, in similar ways for both geometries. However, at the
critical point for which the characteristic exponent is z = 3/2 in one
dimension, the mean-field value z = 2 is approached for very large systems with
constant (finite) aspect ratio. We also treat a second combination of bond (and
boundary) rates where, more typically, sublattice distinction persists. For the
two rate combinations, in continuum or late-time limits, respectively, the
coupled sets of mean-field dynamical equations become tractable with various
techniques and give a two-band spectrum, gapless in the critical phase. While
for the second rate combination quantitative discrepancies between mean-field
theory and simulations increase for most properties and boundary rates
investigated, theory still is qualitatively correct in general, and gives a
fairly good quantitative account of features such as the late-time evolution of
density profile differences from their steady-state values.
