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：近年来图论研究的一个趋势是通过聚焦于用于表示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，几何非均匀随机图（GIRGs）能很好地表示复杂的现实世界网络，其边概率被表达为一个函数，该函数依赖于顶点在底层几何空间中的（异质）权重和距离分布。虽然该模型的大多数参数已得到较好理解，但基础空间的维度如何影响图结构尚不明确。本文通过研究GIRGs结构随维度增加的变化，补充了现有关于几何随机图模型维度的研究以及对现实网络维度判定的持续探索。我们证明在极限情况下，GIRGs趋近于非几何非均匀随机图，并揭示了几何性衰减对重要图结构的影响速度。特别地，我们研究了给定规模团的期望数量以及团数，并刻画了其行为发生根本性变化的相变点。最后，我们的研究结果有助于更深入理解先前关于维度对几何随机图影响的结论。