Pair correlation functions for identifying spatial correlation in discrete domains

TL;DR

This paper introduces new pair correlation functions for discrete domains, using different metrics and tessellations, to better quantify spatial correlations in multi-agent systems.

Contribution

It defines novel PCFs for discrete lattices with various metrics and tessellations, extending their applicability to irregular and complex domains.

Findings

New PCFs for square, hexagonal, triangular, and cuboidal lattices.

Comparison shows improved correlation detection over previous methods.

Applicable to irregular lattices with less intuitive spatial correlation recognition.

Abstract

Identifying and quantifying spatial correlation are important aspects of studying the collective behaviour of multi-agent systems. Pair correlation functions (PCFs) are powerful statistical tools which can provide qualitative and quantitative information about correlation between pairs of agents. Despite the numerous PCFs defined for off-lattice domains, only a few recent studies have considered a PCF for discrete domains. Our work extends the study of spatial correlation in discrete domains by defining a new set of PCFs using two natural and intuitive definitions of distance for a square lattice: the taxicab and uniform metric. We show how these PCFs improve upon previous attempts and compare between the quantitative data acquired. We also extend our definitions of the PCF to other types of regular tessellation which have not been studied before, including hexagonal, triangular and cuboidal. Finally, we provide a comprehensive PCF for any tessellation and metric allowing investigation of spatial correlation in irregular lattices for which recognising correlation is less intuitive.