A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.

翻译：近期图论研究的一个趋势是，通过关注用于表征实际实例的随机图模型，使理论分析更贴近实证观察。研究发现，几何非齐次随机图（GIRGs）通过将边概率表示为依赖于（异质）顶点权重及顶点在底层几何空间中分布距离的函数，能够良好地表征复杂真实世界网络。尽管该模型多数参数已被充分理解，但底层空间维度如何影响图结构仍不明确。本文通过研究GIRGs结构随维度增加的变化规律，补充了关于几何随机图模型维度的现有研究，并深化了正在进行的真实世界网络维度测定工作。我们证明在极限情况下，GIRGs趋近于非几何非齐次随机图，并揭示了几何效应衰减速度对重要图结构的影响机制。特别地，我们研究了给定大小团的期望数量与团数，并刻画了其行为发生根本变化的相变点。最终，本文研究有助于更深入理解此前关于维度对几何随机图影响的相关结论。