  We describe numerical simulations and analyses of a quasi-one-dimensional
(Q1D) model of glassy dynamics. In this model, hard rods undergo Brownian
dynamics through a series of narrow channels connected by $J$ intersections. We
do not allow the rods to turn at the intersections, and thus there is a single,
continuous route through the system. This Q1D model displays caging behavior,
collective particle rearrangements, and rapid growth of the structural
relaxation time, which are also found in supercooled liquids and glasses. The
mean-square displacement $\Sigma(t)$ for this Q1D model displays several
dynamical regimes: 1) short-time diffusion $\Sigma(t) \sim t$, 2) a plateau in
the mean-square displacement caused by caging behavior, 3) single-file
diffusion characterized by anomalous scaling $\Sigma(t) \sim t^{0.5}$ at
intermediate times, and 4) a crossover to long-time diffusion $\Sigma(t) \sim
t$ for times $t$ that grow with the complexity of the circuit. We develop a
general procedure for determining the structural relaxation time $t_D$, beyond
which the system undergoes long-time diffusion, as a function of the packing
fraction $\phi$ and system topology. This procedure involves several steps: 1)
define a set of distinct microstates in configuration space of the system, 2)
construct a directed network of microstates and transitions between them, 3)
identify minimal, closed loops in the network that give rise to structural
relaxation, 4) determine the frequencies of `bottleneck' microstates that
control the slow dynamics and time required to transition out of them, and 5)
use the microstate frequencies and lifetimes to deduce $t_D(\phi)$. We find
that $t_D$ obeys power-law scaling, $t_D\sim (\phi^* - \phi)^{-\alpha}$, where
both $\phi^*$ (signaling complete kinetic arrest) and $\alpha>0$ depend on the
system topology.
