  We investigate the topological properties of dynamical states evolving on
periodic oriented graphs. This evolution, that encodes the scattering processes
occurring at the nodes of the graph, is described by a single-step global
operator, in the spirit of the Ho-Chalker model. When the successive scattering
events follow a cyclic sequence, the corresponding scattering network can be
equivalently described by a discrete time-periodic unitary evolution, in line
with Floquet systems. Such systems may present anomalous topological phases
where all the first Chern numbers are vanishing, but where protected edge
states appear in a finite geometry. To investigate the origin of such anomalous
phases, we introduce the phase rotation symmetry, a generalization of usual
symmetries which only occurs in unitary systems (as opposed to Hamiltonian
systems). Equipped with this new tool, we explore a possible explanation of the
pervasiveness of anomalous phases in scattering network models, and we define
bulk topological invariants suited to both equivalent descriptions of the
network model, which fully capture the topology of the system. We finally show
that the two invariants coincide, again through a phase rotation symmetry
arising from the particular structure of the network model.
