  The concept of Turing instability, namely that diffusion can destabilize the
uniform steady state, is well known either in the context of partial
differential equations (PDEs) or in networks of dynamical systems. Recently
reaction-diffusion equations with cross-diffusion terms have been investigated,
showing an analogous effect called cross-diffusion induced instability. In this
paper, we extend this concept to networks of dynamical systems, showing that
the spectrum of the graph Laplacian determines the instability appearance, as
well as the spectrum of the Laplace operator in reaction-diffusion equations.
We extend to network dynamics a particular network model for competing species,
coming from the PDEs context. In particular, the influence of different
topology structures on the cross-diffusion induced instability is highlighted,
considering regular rings and lattices, and also small-world,
Erd\H{o}s-R\'eyni, and Barab\'asi-Albert networks.
