  We study the failure process of fiber bundles on complex networks focusing on
the effect of the degree of disorder of fibers' strength on the transition from
localized to mean field behaviour. Starting from a regular square lattice we
apply the Watts-Strogatz rewiring technique to introduce long range random
connections in the load transmission network and analyze how the ultimate
strength of the bundle and the statistics of the size of failure cascades
change when the rewiring probability is gradually increased. Our calculations
revealed that the degree of strength disorder of nodes of the network has a
substantial effect on the localized to mean field transition. In particular, we
show that the transition sets on at a finite value of the rewiring probability,
which shifts to higher values as the degree of disorder is reduced. The
transition is limited to a well defined range of disorder, so that there exists
a threshold disorder of nodes' strength below which the randomization of the
network structure does not provide any improvement neither of the overall load
bearing capacity nor of the cascade tolerance of the system. At low strength
disorder the fully random network is the most stable one, while at high
disorder best cascade tolerance is obtained at a lower structural randomness.
Based on the interplay of the network structure and strength disorder we
construct an analytical argument which provides a reasonable description of the
numerical findings.
