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Investigating the topology of interacting networks - Theory and application to coupled climate subnetworks
Jonathan F. Donges, Hanna C. H. Schultz, Norbert Marwan, Yong Zou, Juergen Kurths
TL;DR
The paper addresses the need to analyze systems as interacting networks rather than only as single networks. It develops graph-theoretical measures and applies coupled climate subnetworks to atmospheric geopotential-height data, revealing structured vertical interactions and circulation features. The approach offers a network-based route to studying interacting Earth-system components, with interpretation constrained by the noncausal nature of functional climate networks.
Problem
Existing interacting-network research has emphasized global properties, leaving the roles of individual vertices and subnetworks in pairwise interactions insufficiently characterized.
Method
The paper partitions networks into subnetworks, defines local and global interaction measures, and applies coupled climate subnetwork analysis to atmospheric geopotential-height fields.
Results
The coupled climate subnetwork analysis yields a consistent picture of large-scale atmospheric circulation and indicates nontrivial vertical interaction topology, while cross-betweenness identifies regions mediating vertical wind-field interactions.
Takeaways & Limitations
Coupled climate subnetworks provide a first step toward studying the Earth system and its complex interacting components from a network perspective.
Takeaways & Limitations
Because climate networks are functional networks inferred from statistical dynamics, their topological connections do not directly establish causal relationships between vertices.
Abstract
from arXiv · showhide
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.
1 Introduction
Many systems are better represented as interacting subnetworks, while climate networks provide a basis for studying statistical relationships in climatological fields. The paper develops coupled climate subnetwork analysis to investigate atmospheric vertical dynamics and circulation.
- 1 Introduction: Interacting networks represent systems whose components are themselves subnetworks connected by internal dependencies and cross-subnetwork interactions.Subnets may be naturally defined or obtained through community detection.
- 1 Introduction: Climate networks have revealed statistical relationships in climatological fields, including ENSO-related correlation patterns and enhanced global surface-temperature matter and energy flow.
- 1 Introduction: The paper addresses the challenge of understanding interactions among Earth-system domains by developing coupled climate subnetwork analysis.
- 1 Introduction: The general framework quantifies the roles of individual vertices and subnetworks in interactions between subnetworks across local, mesoscopic, and global scales.
2 Theory: The topology of interacting networks
The framework decomposes a network into subnetworks and introduces local and global measures for quantifying their interaction topology. These measures capture direct connectivity, clustering, interaction efficiency, communication control, separation, and organization.
- 2 Theory: The topology of interacting networks: A network is partitioned into disjoint vertex sets and corresponding internal and cross-edge sets, defining subnetworks and their mutual interactions.
- Local measures: Local measures quantify direct influence, local interdependency organization, interaction efficiency, and communication control between vertices and subnetworks.They include cross-degree, local cross-clustering, cross-closeness, and cross-betweenness.
- Local measures: Cross-degree measures a vertex’s direct connections to a target subnetwork, while local cross-clustering estimates whether its neighbors there are connected.
- Local measures: Cross-closeness measures a vertex’s topological efficiency of interaction with a subnetwork using shortest paths that may traverse any vertices in the full graph.
- Local measures: Cross-betweenness measures a vertex’s role in mediating communication between two subnetworks and can distinguish relational hubs by their vulnerability and redundancy.
- Global measures: Global measures use cross-edge density, cross-clustering or transitivity, and cross-average path length to characterize separation, organization, and interaction efficiency between subnetworks.Low cross-average path length indicates closely interwoven subnetworks, whereas high values indicate greater topological separation.
3 Application: Analysing the vertical dynamical structure of the Earth’s atmosphere
Coupled climate subnetworks reveal structured, direction-dependent interactions between atmospheric isobaric surfaces rather than random connectivity. Their cross-measures identify height-dependent coupling patterns and regional pathways consistent with atmospheric stratification and circulation.
- Approach: Coupled climate subnetworks extend climate-network analysis to dynamical interrelationships between different climatological fields.The approach focuses on statistical interrelationships between atmospheric isobaric surfaces while recognising that functional-network links do not directly establish causality.
- Vertical coupling: Cross-edge density is generally lower than internal upper-surface density, while ρi/ρ1i increases with height, indicating increasing separation between within-surface and cross-surface dynamics.The near-surface internal density is also generally larger than cross-edge density, although the ratio ρ1/ρ1i can reach approximately 0.7.
- Vertical coupling: Cross-edge density has maxima near 1–3 km and 16 km, a minimum near 12 km, and a weaker inversion near 26 km that appears only for small thresholds.These extrema differ from the internal-density extrema because internal and cross-surface measures capture distinct atmospheric processes.
- Vertical coupling: Cross-average path length has minima at 3 km and 16 km and a maximum at 11 km, complementing the cross-edge-density pattern.Lower L1i indicates tighter dynamical relationships, whereas higher values indicate weaker coupling between isobaric surfaces.
- Interaction topology: Directional clustering measures depart from random-connectivity expectations across most heights, with more consistent upward than downward patterns and convergence toward random values above 20 km.The observed nontrivial interaction topology remains evident under a null model preserving cross-degree sequences.
- Regional structure: Cross-degree and cross-closeness centralities are higher in the tropics and polar regions, with northern polar values exceeding Antarctic and Southern Ocean values.Upward measures show tropical maxima near 1 km and 16 km, while downward measures show maxima near 3 km and 16 km plus a northern mid-latitude maximum near 14 km.
4 Conclusions
The paper develops a graph-theoretical framework for analysing interaction topology between subnetworks and applies it to geopotential-height data in the atmosphere. The coupled climate subnetwork approach reveals circulation features and may complement established methods for jointly analysing climate datasets.
- The framework investigates interaction topology between pairs of subnetworks embedded within a network of networks.
- Applied to a four-dimensional geopotential-height dataset, the framework yielded a consistent picture of the Earth’s large-scale atmospheric circulation.
- Cross-betweenness centrality may reveal previously unknown atmospheric features and help address open questions about general circulation.
- Coupled climate subnetworks offer a tool for integrated analysis of multiple climatological fields and can complement established linear methods such as canonical correlation analysis.