Scaling of the clustering function in spatial inhomogeneous random graphs

TL;DR

This paper analyzes how local clustering scales with degree in a broad class of spatial inhomogeneous random graphs, revealing phase transitions and geometric insights into typical triangles across different regimes.

Contribution

It identifies the scaling behavior and leading constants of local clustering in a unified framework for various spatial random graph models, highlighting phase transitions.

Findings

Clustering scaling exhibits phase transitions across regimes.

Leading constants of clustering are explicitly identified.

Geometry of typical triangles varies with the interpolation parameter.

Abstract

We consider an infinite spatial inhomogeneous random graph model with an integrable connection kernel that interpolates nicely between existing spatial random graph models. Key examples are versions of the weight-dependent random connection model, the infinite geometric inhomogeneous random graph, and the age-based random connection model. These infinite models arise as the local limit of the corresponding finite models, see \cite{LWC_SIRGs_2020}. For these models we identify the scaling of the \emph{local clustering} as a function of the degree of the root in different regimes in a unified way. We show that the scaling exhibits phase transitions as the interpolation parameter moves across different regimes. In addition to the scaling we also identify the leading constants of the clustering function. This allows us to draw conclusions on the geometry of a \emph{typical} triangle contributing to the clustering in the different regimes.