In this paper, we study the entropy of a hard random geometric graph (RGG), a commonly used model for spatial networks, where the connectivity is governed by the distances between the nodes. Formally, given a connection range $r$, a hard RGG $G_m$ on $m$ vertices is formed by drawing $m$ random points from a spatial domain, and then connecting any two points with an edge when they are within a distance $r$ from each other. The two domains we consider are the $d$-dimensional unit cube $[0,1]^d$ and the $d$-dimensional unit torus $\mathbb{T}^d$. We derive upper bounds on the entropy $H(G_m)$ for both these domains and for all possible values of $r$. In a few cases, we obtain an exact asymptotic characterization of the entropy by proving a tight lower bound. Our main results are that $H(G_m) \sim dm \log_2m$ for $0 < r \leq 1/4$ in the case of $\mathbb{T}^d$ and that the entropy of a one-dimensional RGG on $[0,1]$ behaves like $m\log m$ for all $0<r<1$. As a consequence, we can infer that the asymptotic structural entropy of an RGG on $\mathbb{T}^d$, which is the entropy of an unlabelled RGG, is $Ω((d-1)m \log_2m)$ for $0 < r \leq 1/4$. For the rest of the cases, we conjecture that the entropy behaves asymptotically as the leading order terms of our derived upper bounds.

翻译：本文研究了硬随机几何图（RGG）的熵，该模型是空间网络的常用模型，其中节点间的连通性由其距离决定。形式上，给定连接范围$r$，在$m$个顶点上的硬RGG $G_m$通过从空间域中抽取$m$个随机点构成，当任意两点间的距离小于等于$r$时，它们之间以一条边相连。我们考虑的两个空间域是$d$维单位立方体$[0,1]^d$和$d$维单位环面$\mathbb{T}^d$。我们针对这两个域以及$r$的所有可能取值，推导了熵$H(G_m)$的上界。在少数情况下，我们通过证明紧的下界，获得了熵的精确渐近刻画。我们的主要结果是：对于$\mathbb{T}^d$，当$0 < r \leq 1/4$时，$H(G_m) \sim dm \log_2m$；而对于定义在$[0,1]$上的一维RGG，其熵对所有$0<r<1$均表现为$m\log m$。由此，我们可以推断，对于$0 < r \leq 1/4$，$\mathbb{T}^d$上RGG的渐近结构熵（即无标号RGG的熵）为$Ω((d-1)m \log_2m)$。对于其余情况，我们推测熵的渐近行为与我们推导出的上界的主阶项一致。