Eur. Phys. J. B (2012) 85: 38
https://doi.org/10.1140/epjb/e2011-20678-7

Regular Article

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

¹ Potsdam Institute for Climate Impact Research, Transdisciplinary Concepts and Methods, P.O. Box 60 12 03, 14412 Potsdam, Germany
² Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany
³ Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
⁴ Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB 24 UE, UK

^a e-mail: yong.zou@pik-potsdam.de

Received: 18 August 2011
Received in final form: 12 December 2011
Published online: 23 January 2012

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.