  We investigate the dynamic phase transition from partially or fully arrested
state to spatiotemporal chaos in coupled logistic maps on a small-world
network. Persistence of local variables in coarse grained sense acts as an
excellent order parameter to study this transition. We investigate the phase
diagram by varying coupling strength and small-world rewiring probability p of
nonlocal connections. The persistent region is a compact region bounded by two
critical lines where band-merging crisis occurs. On one critical line, the
persistent sites shows a nonexponential (stretched exponential) decay for all p
while for another one, it shows crossover from nonexponential to exponential
behavior as p tends to 1. With an effectively antiferromagnetic coupling,
coupling to two neighbors on either side leads to exchange frustration. Apart
from exchange frustration, non-bipartite topology and nonlocal couplings in a
small-world networks could be reason for anomalous relaxation. The distribution
of trap times in asymptotic regime has a long tail as well. The dependence of
temporal evolution of persistence on initial conditions is studied and a
scaling form for persistence after waiting time is proposed. We present a
simple possible model for this behavior.
