  Reconfigurable intelligent surface (RIS) provides a promising way to
proactively augment propagation environments for better transmission
performance in wireless communications. Existing multi-RIS works mainly focus
on link-level optimization with predetermined transmission paths, which cannot
be directly extended to system-level management, since they neither consider
the interference caused by undesired scattering of RISs, nor the performance
balancing between different transmission paths. To address this, we study an
innovative multi-hop multi-RIS communication system, where a base station (BS)
transmits information to a set of distributed users over multi-RIS
configuration space in a multi-hop manner. The signals for each user are
subsequently reflected by the selected RISs via multi-reflection line-of-sight
(LoS) links. To ensure that all users have fair access to the system to avoid
excessive number of RISs serving one user, we aim to find the optimal beam
reflecting path for each user, while judiciously determining the path
scheduling strategies with the corresponding beamforming design to ensure the
fairness. Due to the presence of interference caused by undesired scattering of
RISs, it is highly challenging to solve the formulated multi-RIS multi-path
beamforming optimization problem. To solve it, we first derive the optimal
RISs' phase shifts and the corresponding reflecting path selection for each
user based on its practical deployment location. With the optimized
multi-reflection paths, we obtain a feasible user grouping pattern for
effective interference mitigation by constructing the maximum independent sets
(MISs). Finally, we propose a joint heuristic algorithm to iteratively update
the beamforming vectors and the group scheduling policies to maximize the
minimum equivalent data rate of all users.
