  Traveling fronts and stationary localized patterns in bistable
reaction-diffusion systems have been broadly studied for classical continuous
media and regular lattices. Analogs of such non-equilibrium patterns are also
possible in networks. Here, we consider traveling and stationary patterns in
bistable one-component systems on random Erd\"os-R\'enyi, scale-free and
hierarchical tree networks. As revealed through numerical simulations,
traveling fronts exist in network-organized systems. They represent waves of
transition from one stable state into another, spreading over the entire
network. The fronts can furthermore be pinned, thus forming stationary
structures. While pinning of fronts has previously been considered for chains
of diffusively coupled bistable elements, the network architecture brings about
significant differences. An important role is played by the degree (the number
of connections) of a node. For regular trees with a fixed branching factor, the
pinning conditions are analytically determined. For large Erd\"os-R\'enyi and
scale-free networks, the mean-field theory for stationary patterns is
constructed.

Pattern Formation and Solitons;Dynamical Systems;Adaptation and Self-Organizing Systems