In this work, a novel model of the random geometric graph (RGG), namely the isotropic random geometric graphs (IRGG) has been developed and its topological properties in two dimensions have been studied in details. The defining characteristics of RGG and IRGG are the same — two nodes are connected by an edge if their distance is less than a fixed value, called the connection radius. However, IRGGs have two major differences from regular RGGs. Firstly, the shape of their boundaries — which is circular. It brings very little changes in final results but gives a significant advantage in analytical calculations of the network properties. Secondly, it opens up the possibility of an empty concentric region inside the network. The empty region contains no nodes but allows the communicating edges between the nodes to pass through it. This second difference causes significant alterations in physically relevant network properties such as average degree, connectivity, clustering coefficient and average shortest path. Analytical expressions for most of these features have been provided. These results agree well with those obtained from simulations. Apart from the applicability of the model due to its symmetry and simplicity, the scope of incorporating a penetrable cavity makes it suitable for potential applications in wireless communication networks that often have a node-free region.

在这项工作中，开发了一种新的随机几何图（RGG）模型，即各向同性随机几何图（IRGG），并详细研究了其二维拓扑特性。RGG 和 IRGG 的定义特征是相同的——如果两个节点的距离小于一个固定值，称为连接半径，则它们通过一条边连接。但是，IRGG 与常规 RGG 有两个主要区别。首先，它们的边界的形状——圆形。它对最终结果带来的变化很小，但在网络属性的分析计算中具有显着优势。其次，它开辟了网络内部空的同心区域的可能性。空白区域不包含节点，但允许节点之间的通信边通过它。第二个差异导致物理相关网络属性的显着变化，例如平均度、连通性、聚类系数和平均最短路径。已经提供了大多数这些特征的分析表达式。这些结果与从模拟中获得的结果非常吻合。除了由于其对称性和简单性而具有的适用性之外，结合可穿透腔的范围使其适用于通常具有无节点区域的无线通信网络中的潜在应用。