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

Abstract

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.

Edge anisotropy and the geometric perspective on flow networks

“…systems often needs to be taken into account as well. [17][18][19][20] The latter aspect relates to an entirely different class of spatial network characteristics. 8 At the vertex level, the spatial heterogeneity of vertex positions can be quantified by their local density.…”

On the influence of spatial sampling on climate networks

“…While the effects of the area sizes themselves are discussed in detail in Heitzig et al (2012) and can be removed using their proposed consistently weighted network measures, this study shows that the problem goes further, when considering the underlying physical system. While the node size distribution in the two clustered sampling versions discussed in Sect.…”

“…Spatial sampling effects have been discussed (Heitzig et al, 2012), where they proposed weighted measures to deal with differently sized nodes. In climate networks the related issue of temporal sampling and irregularity are treated in Rehfeld et al (2011) and the role of boundary effects was investigated in Rheinwalt et al (2012).…”

Unified functional network and nonlinear time series analysis for complex systems science: Thepyunicornpackage

“…measure equals its unweighted counterpart. 64 For example, the corrected n.s.i. degree k Ãx v and local clustering coefficient C Ãx v are…”

“…degree and local clustering coefficient to the functional climate network of surface air temperature dynamics in the northern polar region. 64 A ClimateNetwork object net was generated as described in Sec. III B, with nodes placed on a regular latitude-longitude grid on the Earth's surface.…”