# Beyond Diagonal Reconfigurable Intelligent Surfaces Utilizing Graph Theory: Modeling, Architecture Design, and Optimization

arXiv: https://arxiv.org/abs/2305.05013 · HTML: https://mlsanity.com/paper/2305.05013/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2023.

## Built on by (most cited first, 2 of 2)

- [Beyond-Diagonal RIS Under Non-Idealities: Learning-Based Architecture Discovery and Optimization](https://mlsanity.com/paper/2510.15701/) (2025)
- [Quantum Intelligence Meets BD-RIS-Enabled AmBC: Challenges, Opportunities, and Practical Insights](https://mlsanity.com/paper/2512.23400/) (2025)

## Abstract (arXiv)

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.