Structural properties of spatially embedded networks

TL;DR

This paper investigates how spatial constraints influence the structural properties of embedded networks, revealing phase transitions in network behavior depending on the decay parameter of link probability and the embedding space dimension.

Contribution

It identifies three distinct regimes of network structure based on the decay parameter, showing how spatial constraints affect properties like path length and clustering.

Findings

Networks with elta<d are small-worlds with zero clustering.

Intermediate elta regimes show increased path length and non-zero clustering.

For elta>2d, networks become large with high clustering.

Abstract

We study the effects of spatial constraints on the structural properties of networks embedded in one or two dimensional space. When nodes are embedded in space, they have a well defined Euclidean distance $r$ between any pair. We assume that nodes at distance $r$ have a link with probability $p(r) \sim r^{- \delta}$. We study the mean topological distance $l$ and the clustering coefficient $C$ of these networks and find that they both exhibit phase transitions for some critical value of the control parameter $\delta$ depending on the dimensionality $d$ of the embedding space. We have identified three regimes. When $\delta <d$, the networks are not affected at all by the spatial constraints. They are ``small-worlds'' $l\sim \log N$ with zero clustering at the thermodynamic limit. In the intermediate regime $d<\delta<2d$, the networks are affected by the space and the distance increases and becomes a power of $\log N$, and have non-zero clustering. When $\delta>2d$ the networks are ``large'' worlds $l \sim N^{1/d}$ with high clustering. Our results indicate that spatial constrains have a significant impact on the network properties, a fact that should be taken into account when modeling complex networks.