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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.
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Several real-world systems can be represented as multi-layer complex networks, i.e. in terms of a superposition of various graphs, each related to a different mode of connection between nodes. Hence, the definition of proper mathematical quantities aiming at capturing the level of complexity of those systems is required. Various attempts have been made to measure the empirical dependencies between the layers of a multiplex, for both binary and weighted networks. In the simplest case, such dependencies are measured via correlation-based metrics: we show that this is equivalent to the use of completely homogeneous benchmarks specifying only global constraints, such as the total number of links in each layer. However, these approaches do not take into account the heterogeneity in the degree and strength distributions, which are instead a fundamental feature of real-world multiplexes. In this work, we compare the observed dependencies between layers with the expected values obtained from reference models that appropriately control for the observed heterogeneity in the degree and strength distributions. This leads to novel multiplexity measures that we test on different datasets, i.e. the International Trade Network (ITN) and the European Airport Network (EAN). Our findings confirm that the use of homogeneous benchmarks can lead to misleading results, and furthermore highlight the important role played by the distribution of hubs across layers.
Nodes in a complex networked system often engage in more than one type of interactions among them; they form a multiplex network with multiple types of links. In real-world complex systems, a nodes degree for one type of links and that for the other are not randomly distributed but correlated, which we term correlated multiplexity. In this paper we study a simple model of multiplex random networks and demonstrate that the correlated multiplexity can drastically affect the properties of giant component in the network. Specifically, when the degrees of a node for different interactions in a duplex Erdos-Renyi network are maximally correlated, the network contains the giant component for any nonzero link densities. In contrast, when the degrees of a node are maximally anti-correlated, the emergence of giant component is significantly delayed, yet the entire network becomes connected into a single component at a finite link density. We also discuss the mixing patterns and the cases with imperfect correlated multiplexity.
The lack of studying the complex organization of directed network usually limits to the understanding of underlying relationship between network structures and functions. Structural controllability and structural predictability, two seemingly unrelated subjects, are revealed in this paper to be both highly dependent on the critical links previously thought to only be able to influence the number of driver nodes in controllable directed networks. Here, we show that critical links can not only contribute to structural controllability, but they can also have a significant impact on the structural predictability of networks, suggesting the universal pattern of structural reciprocity in directed networks. In addition, it is shown that the fraction and location of critical links have a strong influence on the performance of prediction algorithms. Moreover, these empirical results are interpreted by introducing the link centrality based on corresponding line graphs. This work bridges the gap between the two independent research fields, and it provides indications of developing advanced control strategies and prediction algorithms from a microscopic perspective.
We deduce and discuss the implications of self-similarity for the stability in terms of robustness to failure of multiplexes, depending on interlayer degree correlations. First, we define self-similarity of multiplexes and we illustrate the concept in practice using the configuration model ensemble. Circumscribing robustness to survival of the mutually percolated state, we find a new explanation based on self-similarity both for the observed fragility of interconnected systems of networks and for their robustness to failure when interlayer degree correlations are present. Extending the self-similarity arguments, we show that interlayer degree correlations can change completely the stability properties of self-similar multiplexes, so that they can even recover a zero percolation threshold and a continuous transition in the thermodynamic limit, qualitatively exhibiting thus the ordinary stability attributes of noninteracting networks. We confirm these results with numerical simulations.
We study a two states opinion formation model driven by PageRank node influence and report an extensive numerical study on how PageRank affects collective opinion formations in large-scale empirical directed networks. In our model the opinion of a node can be updated by the sum of its neighbor nodes opinions weighted by the node influence of the neighbor nodes at each step. We consider PageRank probability and its sublinear power as node influence measures and investigate evolution of opinion under various conditions. First, we observe that all networks reach steady state opinion after a certain relaxation time. This time scale is decreasing with the heterogeneity of node influence in the networks. Second, we find that our model shows consensus and non-consensus behavior in steady state depending on types of networks: Web graph, citation network of physics articles, and LiveJournal social network show non-consensus behavior while Wikipedia article network shows consensus behavior. Third, we find that a more heterogeneous influence distribution leads to a more uniform opinion state in the cases of Web graph, Wikipedia, and Livejournal. However, the opposite behavior is observed in the citation network. Finally we identify that a small number of influential nodes can impose their own opinion on significant fraction of other nodes in all considered networks. Our study shows that the effects of heterogeneity of node influence on opinion formation can be significant and suggests further investigations on the interplay between node influence and collective opinion in networks.
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