Recently, beyond diagonal reconfigurable intelligent surface (BD-RIS) has been proposed to generalize conventional RIS. BD-RIS has a scattering matrix that is not restricted to being diagonal and thus brings a performance improvement over conventional RIS. While different BD-RIS architectures have been proposed, it still remains an open problem to develop a systematic approach to design BD-RIS architectures achieving the optimal trade-off between performance and circuit complexity. In this work, we propose novel modeling, architecture design, and optimization for BD-RIS based on graph theory. This graph theoretical modeling allows us to develop two new efficient BD-RIS architectures, denoted as tree-connected and forest-connected RIS. Tree-connected RIS, whose corresponding graph is a tree, is proven to be the least complex BD-RIS architecture able to achieve the performance upper bound in multiple-input single-output (MISO) systems. Besides, forest-connected RIS allows us to strike a balance between performance and complexity, further decreasing the complexity over tree-connected RIS. To optimize tree-connected RIS, we derive a closed-form global optimal solution, while forest-connected RIS is optimized through a low-complexity iterative algorithm. Numerical results confirm that tree-connected (resp. forest-connected) RIS achieves the same performance as fully-connected (resp. group-connected) RIS, while reducing the complexity by up to 16.4 times.

翻译：近期，超越对角可重构智能表面（BD-RIS）被提出以推广传统RIS。BD-RIS的散射矩阵不受限于对角形式，因而相较于传统RIS带来了性能提升。尽管已有多种BD-RIS架构被提出，如何系统性地设计出实现性能与电路复杂度最优折中的BD-RIS架构仍是一个开放问题。本文提出基于图论的BD-RIS新型建模方法、架构设计与优化方案。这种图论建模使我们能够开发两种新型高效BD-RIS架构，分别称为树连接RIS和森林连接RIS。树连接RIS（其对应图为树）被证明是能够达到多输入单输出（MISO）系统性能上界的最简BD-RIS架构。此外，森林连接RIS能够在性能与复杂度之间取得平衡，进一步降低树连接RIS的复杂度。针对树连接RIS的优化，我们推导出闭合形式的全局最优解；而森林连接RIS则通过低复杂度迭代算法进行优化。数值结果证实，树连接RIS（森林连接RIS）在实现与全连接RIS（组连接RIS）相同性能的同时，可将复杂度降低高达16.4倍。