  Dynamical patterns in complex networks of coupled oscillators are both of
theoretical and practical interest, yet to fully reveal and understand the
interplay between pattern emergence and network structure remains to be an
outstanding problem. A fundamental issue is the effect of network structure on
the stability of the patterns. We address this issue by using the setting where
random links are systematically added to a regular lattice and focusing on the
dynamical evolution of spiral wave patterns. As the network structure deviates
more from the regular topology (so that it becomes increasingly more complex),
the original stable spiral wave pattern can disappear and a different type of
pattern can emerge. Our main findings are the following. (1) Short-distance
links added to a small region containing the spiral tip can have a more
significant effect on the wave pattern than long-distance connections. (2) As
more random links are introduced into the network, distinct pattern transitions
can occur, which include the transition of spiral wave to global
synchronization, to a chimera-like state, and then to a pinned spiral wave. (3)
Around the transitions the network dynamics is highly sensitive to small
variations in the network structure in the sense that the addition of even a
single link can change the pattern from one type to another. These findings
provide insights into the pattern dynamics in complex networks, a problem that
is relevant to many physical, chemical, and biological systems.
