Sublinear Cuts are the Exception in BDF-GIRGs

TL;DR

This paper introduces BDF-GIRGs, a flexible class of geometric random graphs that model complex social networks more realistically, and classifies when small graph separators exist within these models.

Contribution

It extends GIRGs by incorporating arbitrary hierarchies of coordinate distances, providing a comprehensive classification of small separator existence in these models.

Findings

Small separators exist in BDF-GIRGs under certain conditions.

BDF-GIRGs satisfy a stochastic triangle inequality, indicating clustering.

The models better capture complex social tie formations.

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 Todorovi\'{c}, 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.