From brain to earth and climate systems: Small-world interaction networks or not?

Abstract

We consider recent reports on small-world topologies of interaction networks derived from the dynamics of spatially extended systems that are investigated in diverse scientific fields such as neurosciences, geophysics, or meteorology. With numerical simulations that mimic typical experimental situations we have identified an important constraint when characterizing such networks: indications of a small-world topology can be expected solely due to the spatial sampling of the system along with commonly used time series analysis based approaches to network characterization. The complex dynamics of spatially extended natural systems such as the human brain or the climate and the earth system is notoriously difficult to understand and has repeatedly stimulated a variety of scientific efforts, among them the development of sophisticated data analysis methods. Concepts from network theory promise to improve our understanding of such complex systems. These approaches consider a system as being composed of dynamically interacting subsystems whose functional interdependencies are reflected as links in an interaction network. Interaction networks derived from field data of different dynamical systems have been consistently reported to possess small-world characteristicshigh local clustering and small average shortest path length -which is considered as an indication of a common organization principle of natural dynamical systems. However, little attention has so far been paid to the conditions and assumptions underlying such analysis approaches. We here demonstrate that indications of smallworld characteristics of interaction networks can solely be expected due to the spatial sampling of dynamical systems together with commonly used time series analysis techniques for the characterization of network links. Our findings not only call for the development and use of methods taking into account the spatial sampling of the studied systems but also for a careful interpretation and re-consideration of analysis results obtained so far.

Edge anisotropy and the geometric perspective on flow networks

“…Specifically, it is known that many classical network properties (focusing exclusively on the mutual linkage between vertices) are strongly predetermined by the spatial positions of vertices and edges. 9,19 Examples for this phenomenon include climate networks 43,49 and brain networks. 9 Taking this additional information into account, a more holistic picture of the system's structural organization can be drawn.…”

“…[1][2][3][4] In a growing number of studies, the analyzed networks have been embedded in some physical space, 5 which implies that their vertices take well-defined positions and edges describe physical connections (or, more generally, interdependencies) within this space. Examples for such spatial networks can be found in diverse fields such as infrastructures (e.g., road networks, power grids), [6][7][8] neuronal (brain) networks, 9 or network representations of the dynamical similarity between climate variations observed at distant points on the globe commonly referred to as (functional) climate networks. [10][11][12][13][14][15][16] Due to their embedding in some metric space, spatial networks are not completely described by their topological characteristics.…”

Robust detection of dynamic community structure in networks

“…In many fields, determining the definition of nodes and edges is itself an active area of investigation. 37 See, for example, several recent papers that address such questions in the context of large-scale human brain networks [38][39][40][41][42][43] and in networks more generally. 44 Another important issue is whether to examine a given adjacency matrix in an exploratory manner or to impose structure on it based on a priori knowledge.…”

Data-driven prediction and prevention of extreme events in a spatially extended excitable system

“…Recent years have seen progress to this respect [49] yielding methods differing according to the influence which can be exerted on the system: If the system can be driven, driving-response based methods (see, e.g., Ref. [50]) may be appropriate, whereas correlation-based methods, which are likely affected by the spatial [51] and temporal [52] sampling of the dynamics, may be considered for systems whose dynamics can be only observed but not manipulated in a controlled way (see, e.g., Ref. [53]).…”