Sublinear Cuts are the Exception in BDF-GIRGs
Marc Kaufmann, Raghu Raman Ravi, Ulysse Schaller
TL;DR
This paper broadens Geometric Inhomogeneous Random Graphs (GIRGs) by introducing Boolean Distance Functions (BDFs) on the $d$-torus, enabling hierarchical and feature-driven proximity in network modeling. It classifies the existence of sublinear separators in BDF-GIRGs: if the distance κ is Single-Coordinate Outer-Max (SCOM), the giant component has a sublinear separator of size $o(n)$ along a singled-out coordinate; if κ is non-SCOM, such sublinear separators do not exist whp. A two-round vertex-coordinate exposure technique is developed to extend existing MCD-GIRG arguments to the full BDF family, with edge-insertion criteria adapted to the BDF structure. Additionally, the work proves that all BDF-GIRGs satisfy a stochastic triangle inequality, yielding constant clustering $ extsc{cc}(G)=\Theta(1)$ whp, which aligns with observed clustering in real networks. Overall, the paper delivers a complete classification of sublinear separators for BDF-GIRGs and establishes clustering, significantly broadening the modeling toolkit for realistic network structure.
Abstract
The introduction of geometry has proven instrumental in the efforts towards more realistic models for real-world networks. In Geometric Inhomogeneous Random Graphs (GIRGs), Euclidean Geometry induces clustering of the vertices, which is widely observed in networks in the wild. Euclidean Geometry in multiple dimensions however restricts proximity of vertices to those cases where vertices are close in each coordinate. We introduce a large class of GIRG extensions, called BDF-GIRGs, which capture arbitrary hierarchies of the coordinates within the distance function of the vertex feature space. These distance functions have the potential to allow more realistic modeling of the complex formation of social ties in real-world networks, where similarities between people lead to connections. Here, similarity with respect to certain features, such as familial kinship or a shared workplace, suffices for the formation of ties. It is known that - while many key properties of GIRGs, such as log-log average distance and sparsity, are independent of the distance function - the Euclidean metric induces small separators, i.e. sublinear cuts of the unique giant component in GIRGs, whereas no such sublinear separators exist under the component-wise minimum distance. Building on work of Lengler and Todorović, we give a complete classification for the existence of small separators in BDF-GIRGs. We further show that BDF-GIRGs all fulfill a stochastic triangle inequality and thus also exhibit clustering.