  Robustness against disorder and defects is a pivotal advantage of topological
systems, manifested by absence of electronic backscattering in the quantum Hall
and spin-Hall effects, and unidirectional waveguiding in their classical
analogs. Two-dimensional (2D) topological insulators, in particular, provide
unprecedented opportunities in a variety of fields due to their compact planar
geometries compatible with the fabrication technologies used in modern
electronics and photonics. Among all 2D topological phases, Chern insulators
are to date the most reliable designs due to the genuine backscattering
immunity of their non-reciprocal edge modes, brought via time-reversal symmetry
breaking. Yet, such resistance to fabrication tolerances is limited to
fluctuations of the same order of magnitude as their band gap, limiting their
resilience to small perturbations only. Here, we tackle this vexing problem by
introducing the concept of anomalous non-reciprocal topological networks, that
survive disorder levels with strengths arbitrarily larger than their bandgap.
We explore the general conditions to obtain such unusual effect in systems made
of unitary three-port scattering matrices connected by phase links, and
establish the superior robustness of the anomalous edge modes over the Chern
ones to phase link disorder of arbitrarily large values. We confirm
experimentally the exceptional resilience of the anomalous phase, and
demonstrate its operation by building an ideal anomalous topological circulator
despite its arbitrary shape and large number of ports. Our results pave the way
to efficient, arbitrary planar energy transport on 2D substrates for wave
devices with full protection against large fabrication flaws or imperfections.
