  In this paper, the Turing instability in reaction-diffusion models defined on
complex networks is studied. Here, we focus on three types of models which
generate complex networks, i.e. the Erd\H{o}s-R\'enyi, the Watts-Strogatz, and
the threshold network models. From analysis of the Laplacian matrices of graphs
generated by these models, we numerically reveal that stable and unstable
regions of a homogeneous steady state on the parameter space of two diffusion
coefficients completely differ, depending on the network architecture. In
addition, we theoretically discuss the stable and unstable regions in the cases
of regular enhanced ring lattices which include regular circles, and networks
generated by the threshold network model when the number of vertices is large
enough.
