Growing Random Geometric Graph Models of Super-linear Scaling Law

TL;DR

This paper introduces growing random geometric graph models to explain super-linear scaling laws in complex systems, capturing phenomena like city growth and network properties through a geometric space framework.

Contribution

It proposes a novel geometric network model that reproduces super-linear growth and related complex network phenomena, linking the growth exponent to the space dimension.

Findings

Model reproduces super-linear growth law

Captures scale-free and clustering properties

Links growth exponent to geometric dimension

Abstract

Recent researches on complex systems highlighted the so-called super-linear growth phenomenon. As the system size $P$ measured as population in cities or active users in online communities increases, the total activities $X$ measured as GDP or number of new patents, crimes in cities generated by these people also increases but in a faster rate. This accelerating growth phenomenon can be well described by a super-linear power law $X \propto P^{\gamma}$($\gamma>1$). However, the explanation on this phenomenon is still lack. In this paper, we propose a modeling framework called growing random geometric models to explain the super-linear relationship. A growing network is constructed on an abstract geometric space. The new coming node can only survive if it just locates on an appropriate place in the space where other nodes exist, then new edges are connected with the adjacent nodes whose number is determined by the density of existing nodes. Thus the total number of edges can grow with the number of nodes in a faster speed exactly following the super-linear power law. The models cannot only reproduce a lot of observed phenomena in complex networks, e.g., scale-free degree distribution and asymptotically size-invariant clustering coefficient, but also resemble the known patterns of cities, such as fractal growing, area-population and diversity-population scaling relations, etc. Strikingly, only one important parameter, the dimension of the geometric space, can really influence the super-linear growth exponent $\gamma$.