Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes

Abstract

Networks with nodes embedded in a metric space have gained increasing interest in recent years. The effects of spatial embedding on the networks' structural characteristics, however, are rarely taken into account when studying their macroscopic properties. Here, we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve global and local statistics associated with the nodes' embedding in a metric space. Comparing the original network's and the resulting surrogates' global characteristics allows to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links. We apply our framework to various real-world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes' spatial embedding. Depending on the actual performance of the proposed null models, the networks are categorized into different classes. Since many real-world complex networks are in fact spatial networks, the proposed approach is relevant for disentangling underlying complex system structure from spatial embedding of nodes in many fields, ranging from social systems over infrastructure and neurophysiology to climatology.

Edge anisotropy and the geometric perspective on flow networks

“…However, one has to bear in mind that an unconstrained randomization may not necessarily provide a reasonable benchmark scenario, while "network surrogates" conserving local or even global topological and/or geometric characteristics (such as degree distribution, local degree, and global or local edge length distributions) can be considerable alternatives. 19 This issue should be addressed in some more detail in future work.…”

“…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.…”

“…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.…”

“…Specifically, we apply this new concept to characterize the local spatial organization of two-dimensional flow systems using spatially embedded functional network representations based on the linear correlations among the fields of scalarvalued time series. We do not aim to discuss the possible limitations of topological network characteristics to describing such flow patterns (or general spatially embedded networks 19 ), but rather to highlight the added value provided by the additional consideration of geometric properties.…”

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

“…Furthermore, three different GeoModels are implemented in pyunicorn which construct random network surrogates of a given network by iteratively rewiring its links under different conditions: (i) GeoModel1 (GeoNetwork.randomly_rewire_ geomodel_I) creates a random network with the same global link-length distribution and degree sequence as the one represented by the respective instance of GeoNetwork, (ii) GeoModel2 (GeoNetwork.randomly_rewire_ geomodel_II) additionally preserves the local link-length distributions for each node, and (iii) GeoModel3 (GeoNetwork.randomly_rewire_geomodel_III) additionally sustains the degree-degree correlations (or assortativity) of the original network. 50 …”