Node-weighted interacting network measures improve the representation of real-world complex systems

Network theory provides a rich toolbox consisting of methods, measures, and models for studying the structure and dynamics of complex systems found in nature, society, or technology. Recently, it has been pointed out that many real-world complex systems are more adequately mapped by networks of interacting or interdependent networks, e.g., a power grid showing interdependency with a communication network. Additionally, in many real-world situations it is reasonable to include node weights into complex network statistics to reflect the varying size or importance of subsystems that are represented by nodes in the network of interest. E.g., nodes can represent vastly different surface area in climate networks, volume in brain networks or economic capacity in trade networks. In this letter, combining both ideas, we derive a novel class of statistical measures for analysing the structure of networks of interacting networks with heterogeneous node weights. Using a prototypical spatial network model, we show that the newly introduced node-weighted interacting network measures indeed provide an improved representation of the underlying system’s properties as compared to their unweighted analogues. We apply our method to study the complex network structure of cross-boundary trade between European Union (EU) and non-EU countries finding that it provides important information on trade balance and economic robustness. Introduction. – Complex network theory has been shown to be a powerful tool for analysing the structure and function of many complex systems in nature, society, and technology. Various kinds of measures have been defined recently, mostly based on counting nodes, paths, links or triangles in a network [1–3]. The field of application is wide-spread, e.g., considering social [4], trade [5], biological [6, 7], communication [8] and climate networks [9–13]. When applying network theory to real-world networks, it is often not sufficient to describe the underlying complex system by an isolated network. Instead, a network of interacting networks may provide an improved representation as was shown, e.g., in an analysis of the interdependency between the Internet network and the Italian power grid (a)E-mail: [email protected] (b)E-mail: [email protected] during a blackout in 2008 [14]. In general, interdependent networks behave much differently from isolated ones in terms of robustness to random failure, expected network properties [15, 16] as well as synchronisation behaviour [17]. In order to quantify the structure of interdependent networks, the recently introduced interacting networks approach compromises a set of cross-network measures in analogy to the canonical network measures designed for isolated networks [12, 18, 19]. The method has been applied successfully for analysing the dynamical structure of the lower atmosphere by constructing coupled climate networks from pairs of geopotential height fields at different isobaric surfaces [12,18]. When applying network theory to real-world problems, nodes need not all bear the same importance for the network’s properties, but there may be nodes having a strong p-1 ar X iv :1 30 1. 08 05 v1 [ ph ys ic s. so cph ] 4 J an 2 01 3 M. Wiedermann et al.