Multiplexity and multireciprocity in directed multiplexes

Real-world multi-layer networks feature nontrivial dependencies among links of different layers. Here we argue that, if links are directed, dependencies are twofold. Besides the ordinary tendency of links of different layers to align as the result of `multiplexity', there is also a tendency to anti-align as the result of what we call `multireciprocity', i.e. the fact that links in one layer can be reciprocated by \emph{opposite} links in a different layer. Multireciprocity generalizes the scalar definition of single-layer reciprocity to that of a square matrix involving all pairs of layers. We introduce multiplexity and multireciprocity matrices for both binary and weighted multiplexes and validate their statistical significance against maximum-entropy null models that filter out the effects of node heterogeneity. We then perform a detailed empirical analysis of the World Trade Multiplex (WTM), representing the import-export relationships between world countries in different commodities. We show that the WTM exhibits strong multiplexity and multireciprocity, an effect which is however largely encoded into the degree or strength sequences of individual layers. The residual effects are still significant and allow to classify pairs of commodities according to their tendency to be traded together in the same direction and/or in opposite ones. We also find that the multireciprocity of the WTM is significantly lower than the usual reciprocity measured on the aggregate network. Moreover, layers with low (high) internal reciprocity are embedded within sets of layers with comparably low (high) mutual multireciprocity. This suggests that, in the WTM, reciprocity is inherent to groups of related commodities rather than to individual commodities. We discuss the implications for international trade research focusing on product taxonomies, the product space, and fitness/complexity metrics.Comment: 20 pages, 8 figure

Reconstruction of multiplex networks with correlated layers

The characterization of various properties of real-world systems requires the knowledge of the underlying network of connections among the system's components. Unfortunately, in many situations the complete topology of this network is empirically inaccessible, and one has to resort to probabilistic techniques to infer it from limited information. While network reconstruction methods have reached some degree of maturity in the case of single-layer networks (where nodes can be connected only by one type of links), the problem is practically unexplored in the case of multiplex networks, where several interdependent layers, each with a different type of links, coexist. Even the most advanced network reconstruction techniques, if applied to each layer separately, fail in replicating the observed inter-layer dependencies making up the whole coupled multiplex. Here we develop a methodology to reconstruct a class of correlated multiplexes which includes the World Trade Multiplex as a specific example we study in detail. Our method starts from any reconstruction model that successfully reproduces some desired marginal properties, including node strengths and/or node degrees, of each layer separately. It then introduces the minimal dependency structure required to replicate an additional set of higher-order properties that quantify the portion of each node's degree and each node's strength that is shared and/or reciprocated across pairs of layers. These properties are found to provide empirically robust measures of inter-layer coupling. Our method allows joint multi-layer connection probabilities to be reliably reconstructed from marginal ones, effectively bridging the gap between single-layer properties and truly multiplex information

On metrics and models for multiplex networks

In this thesis, we extend the concept of null models as canonical ensembles of multi-graphs with given constraints and present new metrics able to characterize real-world layered systems based on their correlation patterns. We make extensive use of the maximum-entropy method in order to find the analytical expression of the expectation values of several topological quantities; furthermore, we employ the maximum-likelihood method to fit the models to real datasets. One of the main contributions of the present work is providing models and metrics that can be directly applied to real data. We introduce improved measures of overlap between layers of a multiplex and exploit such quantities to provide a new network reconstruction method applicable to multi-layer graphs. It turns out that this methodology, applicable to a specific class of multi-layer networks, can be successfully employed to reconstruct the World Trade Multiplex. Furthermore, we illustrate that the maximum-entropy models also allow us to find the so-called backbone of a real network, i.e. the information which is irreducible to the single-node properties and is therefore peculiar to the network itself. We conclude the thesis moving our attention to a different dataset, namely the scientific publication system.Theoretical Physic

Energy and Environmental Flows:Do Most Financialised Countries within the Mediterranean Area Export Unsustainability?

The literature dedicated to the problems of transboundary pollution often aims to verify what the environmental and energy interactions between countries are. Little attention is paid to the financial relations of the phenomenon. We analyze how financial, environmental and energy flows have been redistributed within the main Mediterranean countries, with particular reference to pollution. Applying advanced methods of correlation, we verify the dynamics of transfer processes with the aim of assessing whether the link between economic and financial and environmental flows might support the hypothesis that rich countries export environmental emissions to poor ones. Our results show that richer countries have a significant propensity to export energy, financial flows and polluting emissions. The imbalance is even greater for emissions with local impact. This process is accompanied by a substantial increase in the financial activities of the North Mediterranean countries to the detriment of those of the South, which progressively increase their indebtedness. We find out that the economic and financial development of the North Med is accompanied by an increasing environmental impact measured by the various types of emissions covered by our study. The research shows how the most industrialized countries of the Mediterranean area are increasing the economic and financial gap with respect to the Southern Mediterranean countries

Quid pro quo?: The benefit of reciprocity, multiplexity, and multireciprocity in early career peer support

In the early career phase of higher education, the social relationships with peers are a critical source of developmental support. Peer support relationships tend to be reciprocal and multiplex, such that the actors of the relationships both give and receive multiple facets of developmental support from one another. However, reciprocity and multiplexity alone cannot cover ties that are anti-aligned across the layers of the multiplex network (i.e., one type of support received, and another type of support given in exchange). Therefore, the goal of this study is to integrate reciprocity and multiplexity in order to give consideration to such real-world multilayer relationships. We transferred the approach on multireciprocity introduced by Gemmetto et al. (Phys Rev E 94: 042316, 2016) to weighted network data of 61 university students and explored the possible beneficial effect of reciprocity, multiplexity, and multireciprocity in terms of career outcomes (i.e., objective performance, subjective performance, satisfaction with life). Results revealed no general benefit of mutuality and balance in support relationships. Rather, positive effects emerged for specific constellations of mutuality and support types. Career support in combination with socioemotional support showed to be particularly relevant for early career factors

Interactions between financial and environmental networks in OECD countries

We analyse a multiplex of networks between OECD countries during the decade 2002-2010, which consists of five financial layers, given by foreign direct investment, equity securities, short-term, long-term and total debt securities, and five environmental layers, given by emissions of N O x, P M 10 SO 2, CO 2 equivalent and the water footprint associated with international trade. We present a new measure of cross-layer correlations between flows in different layers based on reciprocity. For the assessment of results, we implement a null model for this measure based on the exponential random graph theory. We find that short-term financial flows are more correlated with environmental flows than long-term investments. Moreover, the correlations between reverse financial and environmental flows (i.e. flows of different layers going in opposite directions) are generally stronger than correlations between synergic flows (flows going in the same direction). This suggests a trade-off between financial and environmental layers, where, more financialised countries display higher correlations between outgoing financial flows and incoming environmental flows from lower financialised countries, which could have important policy implications. Five countries are identified as hubs in this finance-environment multiplex: The United States, France, Germany, Belgium-Luxembourg and the United Kingdom.Comment: Supplementary Information provide