  The advances in understanding complex networks have generated increasing
interest in dynamical processes occurring on them. Pattern formation in
activator-inhibitor systems has been studied in networks, revealing differences
from the classical continuous media. Here we study pattern formation in a new
framework, namely multiplex networks. These are systems where activator and
inhibitor species occupy separate nodes in different layers. Species react
across layers but diffuse only within their own layer of distinct network
topology. This multiplicity generates heterogeneous patterns with significant
differences from those observed in single-layer networks. Remarkably,
diffusion-induced instability can occur even if the two species have the same
mobility rates; condition which can never destabilize single-layer networks.
The instability condition is revealed using perturbation theory and expressed
by a combination of degrees in the different layers. Our theory demonstrates
that the existence of such topology-driven instabilities is generic in
multiplex networks, providing a new mechanism of pattern formation.
