Multiplexity versus correlation: the role of local constraints in real multiplexes

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.Comment: 32 pages, 6 figure

Ground truth? Concept-based communities versus the external classification of physics manuscripts

Community detection techniques are widely used to infer hidden structures within interconnected systems. Despite demonstrating high accuracy on benchmarks, they reproduce the external classification for many real-world systems with a significant level of discrepancy. A widely accepted reason behind such outcome is the unavoidable loss of non-topological information (such as node attributes) encountered when the original complex system is represented as a network. In this article we emphasize that the observed discrepancies may also be caused by a different reason: the external classification itself. For this end we use scientific publication data which i) exhibit a well defined modular structure and ii) hold an expert-made classification of research articles. Having represented the articles and the extracted scientific concepts both as a bipartite network and as its unipartite projection, we applied modularity optimization to uncover the inner thematic structure. The resulting clusters are shown to partly reflect the author-made classification, although some significant discrepancies are observed. A detailed analysis of these discrepancies shows that they carry essential information about the system, mainly related to the use of similar techniques and methods across different (sub)disciplines, that is otherwise omitted when only the external classification is considered.Comment: 15 pages, 2 figure

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

Irreducible network backbones: unbiased graph filtering via maximum entropy

Networks provide an informative, yet non-redundant description of complex systems only if links represent truly dyadic relationships that cannot be directly traced back to node-specific properties such as size, importance, or coordinates in some embedding space. In any real-world network, some links may be reducible, and others irreducible, to such local properties. This dichotomy persists despite the steady increase in data availability and resolution, which actually determines an even stronger need for filtering techniques aimed at discerning essential links from non-essential ones. Here we introduce a rigorous method that, for any desired level of statistical significance, outputs the network backbone that is irreducible to the local properties of nodes, i.e. their degrees and strengths. Unlike previous approaches, our method employs an exact maximum-entropy formulation guaranteeing that the filtered network encodes only the links that cannot be inferred from local information. Extensive empirical analysis confirms that this approach uncovers essential backbones that are otherwise hidden amidst many redundant relationships and inaccessible to other methods. For instance, we retrieve the hub-and-spoke skeleton of the US airport network and many specialised patterns of international trade. Being irreducible to local transportation and economic constraints of supply and demand, these backbones single out genuinely higher-order wiring principles

Local phase transitions in a model of multiplex networks with heterogeneous degrees and inter-layer coupling

Multilayer networks represent multiple types of connections between the same set of nodes. Clearly, a multilayer description of a system adds value only if the multiplex does not merely consist of independent layers, i.e. if the inter-layer overlap is nontrivial. On real-world multiplexes, it is expected that the observed overlap may partly result from spurious correlations arising from the heterogeneity of nodes and partly from true interdependencies. However, no rigorous way to disentangle these two effects has been developed. In this paper we introduce an unbiased maximum-entropy model of multiplexes with controllable node degrees and controllable overlap. The model can be mapped to a generalized Ising model where the combination of node heterogeneity and inter-layer coupling leads to the possibility of local phase transitions. In particular, we find that an increased heterogeneity in the network results in different critical points for different pairs of nodes, which in turn leads to local phase transitions that may ultimately increase the overlap. The model allows us to quantify how the overlap can be increased by either increasing the heterogeneity of the network (spurious correlation) or the strength of the inter-layer coupling (true correlation), thereby disentangling the two effects. As an application, we show that the empirical overlap in the International Trade Multiplex is not merely a spurious result of the correlation between node degrees across different layers, but requires a non-zero inter-layer coupling in its modeling

Stochastically timed predicate-based communication primitives for autonomic computing

Predicate-based communication allows components of a system to send messages and requests to ensembles of components that are determined at execution time through the evaluation of a predicate, in a multicast fashion. Predicate-based communication can greatly simplify the programming of autonomous and adaptive systems. We present a stochastically timed extension of the Software Component Ensemble Language (SCEL) that was introduced in previous work. Such an extension raises a number of non-trivial design and formal semantics issues with different options as possible solutions at different levels of abstraction. We discuss four of these options. We provide formal semantics and an illustration of the use of the language modeling a variant of a bike sharing system, together with some preliminary analysis of the system performance