Clusters in randomly-coloured spatial networks

TL;DR

This paper investigates the formation and properties of clusters in spatially-embedded networks with randomly coloured nodes, providing a null-model for assessing cluster significance in real systems.

Contribution

It introduces a mean-field analytical approach to study clusters in large 2D lattices with random colouring, highlighting their size and shape characteristics.

Findings

Random colouring can produce sizable, interesting clusters.

Simple dynamical measures can distinguish cluster properties.

Analytical results match numerical simulations well.

Abstract

The behaviour and functioning of a variety of complex physical and biological systems depend on the spatial organisation of their constituent units, and on the presence and formation of clusters of functionally similar or related individuals. Here we study the properties of clusters in spatially-embedded networks where nodes are coloured according to a given colouring process. This characterisation will allow us to use spatial networks with uniformly-coloured nodes as a null-model against which the importance, relevance, and significance of clusters of related units in a given real-world system can be assessed. We show that even a uniform and uncorrelated random colouring process can generate coloured clusters of substantial size and interesting shapes, which can be distinguished by using some simple dynamical measures, like the average time needed for a random walk to escape from the cluster. We provide a mean-field approach to study the properties of those clusters in large two-dimensional lattices, and we show that the analytical treatment agrees very well with the numerical results.