This Ph.D. thesis unveils the unique topological phenomena occurring in such networks, focusing on the intricate interplay between their Floquet topology, the presence of disorder, and their unitary scattering at microscopic and macroscopic scales. Using theoretical, numerical, and experimental explorations, it uncovers: 1) the robustness of topological phases under various forms of disorder, 2) the physical distinctions between the two possible topologically non-trivial phases in networks, namely the anomalous Floquet insulator (AFI) and the Chern insulator (CI), and 3) a renormalization group method on unitary scattering systems to explain the microscopic origin of robust macroscopic chiral transport.

We start by categorizing the Floquet topological bands, identified in honeycomb lattice scattering networks with broken time-reversal symmetry (TRS), into three distinct phases: AFI, CI, and trivial phases. The evolution of topological features when tuning the degree of disorder is evaluated by various means, including scattering at external ports, band structures, eigenstates of closed networks, and topological invariants.

Based on theoretical and numerical modelling, we uncover the superior robustness of chiral edge transport in the AFI in the presence of strong distributed disorder, imparted either on the network phase-delay links, on its structure, or on the scattering properties of the nodes. This remarkable robustness, connected to the physics of anomalous Floquet Anderson insulators (AFAI), positions AFI as a promising platform for genuine unidirectional topological edge transports robust to disorder or on-purpose reconfiguration.

A significant proportion of the work is dedicated to experimental validations performed on photonic scattering networks at microwave frequencies, confirming the theoretical predictions. These experiments demonstrate the robust nature of topological edge states amid a wide array of disorder, showcasing the practical potential of AFI in real-world scenarios. Moreover, we introduce innovative methods for measuring topological invariants within finite disordered networks. A specific device is designed to implement twisted boundary conditions, enabling the direct observation and measurement of topological properties.

Finally, a unified framework is proposed to capture the topological properties of scattering networks even in the strong disordered regime: a real-space renormalization group (RG) theory, driven by block-scattering transformations of unitary systems, which, unlike traditional approaches, does not rely on the renormalization of Hamiltonians or wave functions. The resulting RG flows, RG phase diagrams, scaling analysis, and critical behavior studies, prove and explain why chiral topological edge states exist even under the strongest available disorder levels, providing useful guidelines for constructing robust topological photonic systems that never localize.

In conclusion, this work establishes bridges between different concepts in condensed matter physics and photonics, from topology to renormalization group, uncovering the unitary topological physics of scattering networks and the important role of disorder. We envision new opportunities for applications of these physical effects in reconfigurable electromagnetic systems for future communication technologies.