Scale-invariant geometric random graphs

TL;DR

This paper introduces a class of growing geometric random graphs that are invariant under rescaling, exhibiting properties like scale-free degree distributions, high clustering, and unique percolation behavior, similar to real-world web graphs.

Contribution

It presents a novel scale-invariant model of geometric random graphs that captures key features of complex networks such as heterogeneity, hubs, and clustering.

Findings

Existence of scale-free and Poisson degree distributions

Presence of hub nodes with random counts

High clustering and unusual percolation behavior

Abstract

We introduce and analyze a class of growing geometric random graphs that are invariant under rescaling of space and time. Directed connections between nodes are drawn according to influence zones that depend on node position in space and time, mimicking the heterogeneity and increased specialization found in growing networks. Through calculations and numerical simulations we explore the consequences of scale-invariance for geometric random graphs generated this way. Our analysis reveals a dichotomy between scale-free and Poisson distributions of in- and out-degree, the existence of a random number of hub nodes, high clustering, and unusual percolation behaviour. These properties are similar to those of empirically observed web graphs.