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Scaling of the clustering function in spatial inhomogeneous random graphs

Remco van der HofstadPim van der HoornNeeladri Maitra
Dec 2022
摘要
We consider an infinite spatial inhomogeneous random graph model with anintegrable connection kernel that interpolates nicely between existing spatialrandom graph models. Key examples are versions of the weight-dependent randomconnection model, the infinite geometric inhomogeneous random graph, and theage-based random connection model. These infinite models arise as the locallimit of the corresponding finite models, see \cite{LWC_SIRGs_2020}. For thesemodels we identify the scaling of the \emph{local clustering} as a function ofthe degree of the root in different regimes in a unified way. We show that thescaling exhibits phase transitions as the interpolation parameter moves acrossdifferent regimes. In addition to the scaling we also identify the leadingconstants of the clustering function. This allows us to draw conclusions on thegeometry of a \emph{typical} triangle contributing to the clustering in thedifferent regimes.
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