In this paper we study the threshold model of \emph{geometric inhomogeneous random graphs} (GIRGs); a generative random graph model that is closely related to \emph{hyperbolic random graphs} (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their \emph{connectivity}, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a \emph{giant} component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability $1 - \exp(-\Omega(n^{(3-\tau)/2}))$ for graph size $n$ and a degree distribution with power-law exponent $\tau \in (2, 3)$. Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs.

翻译：本文研究了《几何非齐次随机图》（GIRGs）的阈值模型；该生成式随机图模型与《双曲随机图》（HRGs）密切相关。已有观察表明，这些模型在结构和算法特性方面能很好地捕捉复杂真实网络的特征。在对其《连通性》（即图中哪些部分是连通的）进行广泛研究后，我们深入理解了在何种条件下会出现包含图常量部分的大分量。尽管现有结果较为技术化且难以应用，本文旨在提供更易理解的证明。同时，我们显著改进了先前已知的概率保证：对于图规模 n 及幂律指数 τ ∈ (2, 3) 的度分布，GIRGs 以概率 1 - exp(-Ω(n^(3-τ)/2)) 包含一个大分量。基于此，我们进一步推导出 GIRGs 某些诱导子图连通性的相关见解。