Abstract
The analysis of small recurrent substructures, so called network motifs, has become a standard tool of complex network science to unveil the design principles underlying the structure of empirical networks. In many natural systems network nodes are associated with an intrinsic property according to which they can be ordered and compared against each other. Here, we expand standard motif analysis to be able to capture the hierarchical structure in such ordered networks. Our new approach is based on the identification of all ordered 3-node substructures and the visualization of their significance profile. We present a technique to calculate the fine grained motif spectrum by resolving the individual members of isomorphism classes (sets of substructures formed by permuting node-order). We apply this technique to computer generated ensembles of ordered networks and to empirical food web data, demonstrating the importance of considering node order for food-web analysis. Our approach may not only be helpful to identify hierarchical patterns in empirical food webs and other natural networks, it may also provide the base for extending motif analysis to other types of multi-layered networks.
Highlights
Standard network characteristics, such as motif spectra, cannot per se capture the structure of networks that have multiple layers of complexity and cannot be represented as a traditional graph
We propose an extension of standard motif analysis for a specific network type, which we denote as directed ordered networks
We develop a general framework for motif analysis in directed ordered networks
Summary
Using the Niche Model to Generate Directed Ordered Networks. In the following, we show that motif spectra of ordered networks can be used to analyze food web data. Analyzing further food-webs of the Adirondack park (results not shown) we find that this is a common pattern: typically the motif spectrum of the niche model deviates most strongly from that of the most abundant substructure in the empirical data. If the body-size order is neglected by summing over all member IDs, the spectrum reduces to the 13 standard unordered motif classes and the spectra of empirical and randomized networks become indistinguishable (Fig. 5c). These results indicate that hierarchy due to body-size is a crucial aspect of the structure of empirical food webs. The precise role of body-size order for structuring natural food webs provides an intriguing possibility for future research
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