With the fast development of reconfigurable intelligent surface (RIS), the network topology becomes more complex and varied, which makes the network design and analysis extremely challenging. Most of the current works adopt the binary system stochastic geometric, missing the coupling relationships between the direct and reflected paths caused by RISs. In this paper, we first define the typical triangle which consists of a base station (BS), a RIS and a user equipment (UE) as the basic ternary network unit in a RIS-assisted ultra-dense network (UDN). In addition, we extend the Campbell's theorem to the ternary system and present the ternary probability generating functional (PGFL) of the stochastic geometry. Based on the ternary stochastic geometry theory, we derive and analyze the coverage probability, area spectral efficiency (ASE), area energy efficiency (AEE) and energy coverage efficiency (ECE) of the RIS-assisted UDN system. Simulation results show that the RISs can improve the system performances, especially for the UE who has a high signal to interference plus noise ratio (SINR), as if the introduced RIS brings in Matthew effect. This phenomenon of RIS is appealing for guiding the design of complex networks.

翻译：随着可重构智能表面（RIS）的快速发展，网络拓扑变得日益复杂多样，给网络设计与分析带来了极大挑战。现有工作大多采用二元系统随机几何模型，忽略了RIS导致的直射路径与反射路径之间的耦合关系。本文首先将基站（BS）、RIS与用户设备（UE）构成的典型三角形定义为RIS辅助超密集网络（UDN）的基本三元网络单元。在此基础上，将坎贝尔定理扩展至三元系统，提出三元随机几何概率生成泛函（PGFL）。基于三元随机几何理论，推导并分析了RIS辅助UDN系统的覆盖概率、区域频谱效率（ASE）、区域能量效率（AEE）及能量覆盖效率（ECE）。仿真结果表明，RIS能够提升系统性能，尤其对具有较高信干噪比（SINR）的UE，呈现类似马太效应的增强现象。RIS这一特性对复杂网络设计具有重要指导意义。