Investigating the topology of interacting networks - Theory and application to coupled climate subnetworks (1102.3067v1)

Abstract: Network theory provides various tools for investigating the structural or functional topology of many complex systems found in nature, technology and society. Nevertheless, it has recently been realised that a considerable number of systems of interest should be treated, more appropriately, as interacting networks or networks of networks. Here we introduce a novel graph-theoretical framework for studying the interaction structure between subnetworks embedded within a complex network of networks. This framework allows us to quantify the structural role of single vertices or whole subnetworks with respect to the interaction of a pair of subnetworks on local, mesoscopic and global topological scales. Climate networks have recently been shown to be a powerful tool for the analysis of climatological data. Applying the general framework for studying interacting networks, we introduce coupled climate subnetworks to represent and investigate the topology of statistical relationships between the fields of distinct climatological variables. Using coupled climate subnetworks to investigate the terrestrial atmosphere's three-dimensional geopotential height field uncovers known as well as interesting novel features of the atmosphere's vertical stratification and general circulation. Specifically, the new measure "cross-betweenness" identifies regions which are particularly important for mediating vertical wind field interactions. The promising results obtained by following the coupled climate subnetwork approach present a first step towards an improved understanding of the Earth system and its complex interacting components from a network perspective.

• The paper introduces a graph-theoretical framework to quantify interactions within and between subnetworks, applying it to coupled climate subnetworks to analyze atmospheric dynamics.
• The study extends classical network measures like centrality to cross-network analysis, revealing asymmetrical vertical coupling and critical mediation roles between atmospheric layers.
• Findings provide novel insights for climatology and other disciplines modeling complex systems as interacting networks.

Investigating the Topology of Interacting Networks: Applications on Climate Subnetworks

This paper introduces a graph-theoretical framework designed to analyze the interaction structures within complex networks composed of interacting subnetworks. Applying this framework to climate science, the authors offer novel insights into the Earth's atmosphere stratification and circulation dynamics using coupled climate subnetworks.

Graph-theoretical Framework

The paper sets forth a comprehensive methodology to quantify interactions within and between subnetworks, positing that many natural systems can be modeled as networks of networks. Mathematical formulations are provided to extend classic measures — such as degree centrality, closeness centrality, and betweenness centrality — to the cross-network context, addressing both local and global measures. This approach facilitates a nuanced analysis of how vertices or subnetworks influence one another and is aimed at assessing the functional roles of elements within complex interacting networks.

Application to Climate Networks

The framework is applied to investigate the geopotential height field. Constructing coupled climate subnetworks allows for the examination of statistical relationships between various isobaric surfaces representing distinct layers of the atmosphere. By analyzing the Pearson correlations of atmospheric geopotential height anomaly time series, coupled climate subnetworks uncover interesting patterns of atmospheric circulatory behavior and stratification.

Key Findings

• Cross-edge Density and Cross-average Path Length:
  • A pivotal discovery is the identification of notable elevations in cross-edge density between the near-surface layer and higher atmospheric layers at specific altitudes (1-3 km, 16 km), complemented by corresponding reductions at other stratifications (11 km). These inversions suggest strong vertical dynamical interdependencies influenced by processes such as convection and turbulent mixing.
• Cross-degree and Cross-closeness Centralities:
  • Horizontal cross-network measures indicate marked asymmetry between hemispheres. Tropical dynamics show enhanced vertical coupling facilitated by penetrative convection, whereas polar vortices predominantly characterizing the northern hemisphere are reflected in these centralities.
• Cross-betweenness Centrality:
  • Importantly, cross-betweenness centrality highlights regions fulfilling critical mediation roles between vertical layers, suggesting previously overlooked influences on atmospheric circulation patterns.

Theoretical and Practical Implications

This paper's findings have significant implications for climatology, offering alternative means to explore atmospheric dynamics beyond established paradigms. By revealing intricate interaction patterns potentially obscured in conventional analyses, this framework has the potential to improve understanding of climatic shifts and responses to anthropogenic influences.

Furthermore, the broader framework is applicable to various disciplines where complex systems are best understood through interactive contexts, promising developments in fields ranging from neuroscience to social science.

Future Directions

The paper concludes by suggesting potential for extending this analysis to diverse climatological fields, as well as refining network construction methodologies for even richer insights. As large-scale environmental changes challenge existing paradigms, such integrative methods promise invaluable insights into the Earth’s complex systemic interdependencies. There is also room to explore weighted edges and nonlinear dependencies to further refine this approach. Implementing sophisticated domain-specific network null models might prove fruitful for extracting even deeper insights into interacting network structures.

Overall, this paper sets the stage for a new way of understanding complex systems through state-of-the-art network analysis, offering a blueprint for expanded interdisciplinary research using similar methodologies.