Transitions in spatial networks

TL;DR

This paper reviews various types of structural transitions in spatial networks, including topological, localization, and congestion effects, highlighting their significance in real-world applications.

Contribution

It provides a comprehensive overview of recent results on different transitions in spatial networks, emphasizing the impact of space on network behavior.

Findings

Spatial networks exhibit diverse transition behaviors.

Localization transitions affect shortest path patterns.

Congestion influences optimal network structures.

Abstract

Networks embedded in space can display all sorts of transitions when their structure is modified. The nature of these transitions (and in some cases crossovers) can differ from the usual appearance of a giant component as observed for the Erdos-Renyi graph, and spatial networks display a large variety of behaviors. We will discuss here some (mostly recent) results about topological transitions, `localization' transitions seen in the shortest paths pattern, and also about the effect of congestion and fluctuations on the structure of optimal networks. The importance of spatial networks in real-world applications makes these transitions very relevant and this review is meant as a step towards a deeper understanding of the effect of space on network structures.