Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes

TL;DR

This paper introduces node-weighted measures for complex networks to address biases caused by heterogeneous node sizes, ensuring more accurate analysis of spatially embedded and sampled networks across various fields.

Contribution

It proposes an axiomatic, node splitting invariant framework for deriving weighted network measures, improving the analysis of complex systems with heterogeneous node representations.

Findings

Weighted measures reduce bias in network statistics

Applicable to spatially embedded and sampled networks

Enhances analysis in neuroscience and climatology

Abstract

When network and graph theory are used in the study of complex systems, a typically finite set of nodes of the network under consideration is frequently either explicitly or implicitly considered representative of a much larger finite or infinite region or set of objects of interest. The selection procedure, e.g., formation of a subset or some kind of discretization or aggregation, typically results in individual nodes of the studied network representing quite differently sized parts of the domain of interest. This heterogeneity may induce substantial bias and artifacts in derived network statistics. To avoid this bias, we propose an axiomatic scheme based on the idea of node splitting invariance to derive consistently weighted variants of various commonly used statistical network measures. The practical relevance and applicability of our approach is demonstrated for a number of example networks from different fields of research, and is shown to be of fundamental importance in particular in the study of spatially embedded functional networks derived from time series as studied in, e.g., neuroscience and climatology.