Multiplexity and multireciprocity in directed multiplexes

Gemmetto, Valerio; Squartini, Tiziano; Picciolo, Francesco; Ruzzenenti, Franco; Garlaschelli, Diego
In the last few years, the study of multi-layer complex networks has received significant attention. In this work, we provide new measures to analyse dependencies between directed links in different layers of multiplex networks. We show that this requires more than a straightforward extension of the corresponding multiplexity measures that have been developed for undirected multiplexes. In particular, one should take into account the effects of reciprocity, i.e. the tendency of pairs of vertices to establish mutual connections. It is well known that reciprocity is a crucial property of many directed single-layer networks, affecting several dynamical processes taking place on such systems. Here we extend this quantity to directed multiplexes and introduce the notion of multireciprocity, defined as the tendency of links in one layer to be reciprocated by links in a different layer. We introduce multireciprocity measures valid for both binary and weighted networks and then validate these novel quantities on the World Trade Multiplex (WTM), representing the import-export bilateral relations between world countries in different commodities. We show that several pairs of layers exhibit strong multiplexity, an effect which can however be largely encoded into the degree or strength sequences of the various layers. Moreover, we find that most of the pairs of commodities are characterised by positive multireciprocities, even though such values are significantly lower than the usual reciprocity measured on the aggregated network. These results confirm that a multiplex approach to the phenomenon of reciprocity conveys much more information than the analysis of the aggregated network does.
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