Central loops in random planar graphs

TL;DR

This paper investigates the structure of random planar graphs, focusing on central loops and betweenness centrality, and introduces a toy model to understand the emergence of non-trivial patterns like central loops.

Contribution

It presents a simple star-based toy model to analyze the conditions under which central loops become prominent in random planar graphs.

Findings

Loop centrality depends on link weight and position.

Optimal loop position scales with branch number and weight.

Radial branch structure influences loop centrality.

Abstract

Random planar graphs appear in a variety of context and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display non-trivial patterns, such as central loops. We first discuss empirical results for different random planar graphs and we then propose a toy model which allows us to discuss the condition for the emergence of non-trivial patterns such as central loops. This toy model is made of a star network with $N_b$ branches of size $n$ and links of weight $1$, superimposed to a loop at distance $\ell$ from the center and with links of weight $w$. We estimate for this model the BC at the center and on the loop and we show that the loop can be more central than the origin if $w<w_c$ where the threshold of this transition scales as $w_c\sim n/N_b$. In this regime, there is an optimal position of the loop that scales as $\ell_{opt}\sim N_b w/4$. This simple model sheds some light on the organization of these random structures and allows us to discuss the effect of randomness on the centrality of loops. In particular, it suggests that the number and the spatial extension of radial branches are the crucial ingredients that control the existence of central loops.