  We present a case study of how topology can affect synchronization.
Specifically, we consider arrays of phase oscillators coupled in a ring or a
chain topology. Each ring is perfectly matched to a chain with the same initial
conditions and the same random natural frequencies. The only difference is
their boundary conditions: periodic for a ring, and open for a chain. For both
topologies, stable phase-locked states exist if and only if the spread or
"width" of the natural frequencies is smaller than a critical value called the
locking threshold (which depends on the boundary conditions and the particular
realization of the frequencies). The central question is whether a ring
synchronizes more readily than a chain. We show that it usually does, but not
always. Rigorous bounds are derived for the ratio between the locking
thresholds of a ring and its matched chain, for a variant of the Kuramoto model
that also includes a wider family of models.
