13,651 research outputs found
Weighted Multiplex Networks
One of the most important challenges in network science is to quantify the
information encoded in complex network structures. Disentangling randomness
from organizational principles is even more demanding when networks have a
multiplex nature. Multiplex networks are multilayer systems of nodes that
can be linked in multiple interacting and co-evolving layers. In these
networks, relevant information might not be captured if the single layers were
analyzed separately. Here we demonstrate that such partial analysis of layers
fails to capture significant correlations between weights and topology of
complex multiplex networks. To this end, we study two weighted multiplex
co-authorship and citation networks involving the authors included in the
American Physical Society. We show that in these networks weights are strongly
correlated with multiplex structure, and provide empirical evidence in favor of
the advantage of studying weighted measures of multiplex networks, such as
multistrength and the inverse multiparticipation ratio. Finally, we introduce a
theoretical framework based on the entropy of multiplex ensembles to quantify
the information stored in multiplex networks that would remain undetected if
the single layers were analyzed in isolation.Comment: (22 pages, 10 figures
Structure of Triadic Relations in Multiplex Networks
Recent advances in the study of networked systems have highlighted that our
interconnected world is composed of networks that are coupled to each other
through different "layers" that each represent one of many possible subsystems
or types of interactions. Nevertheless, it is traditional to aggregate
multilayer networks into a single weighted network in order to take advantage
of existing tools. This is admittedly convenient, but it is also extremely
problematic, as important information can be lost as a result. It is therefore
important to develop multilayer generalizations of network concepts. In this
paper, we analyze triadic relations and generalize the idea of transitivity to
multiplex networks. By focusing on triadic relations, which yield the simplest
type of transitivity, we generalize the concept and computation of clustering
coefficients to multiplex networks. We show how the layered structure of such
networks introduces a new degree of freedom that has a fundamental effect on
transitivity. We compute multiplex clustering coefficients for several real
multiplex networks and illustrate why one must take great care when
generalizing standard network concepts to multiplex networks. We also derive
analytical expressions for our clustering coefficients for ensemble averages of
networks in a family of random multiplex networks. Our analysis illustrates
that social networks have a strong tendency to promote redundancy by closing
triads at every layer and that they thereby have a different type of multiplex
transitivity from transportation networks, which do not exhibit such a
tendency. These insights are invisible if one only studies aggregated networks.Comment: Main text + Supplementary Material included in a single file.
Published in New Journal of Physic
Centralities of Nodes and Influences of Layers in Large Multiplex Networks
(12 pages, 7 figures)(12 pages, 7 figures)(12 pages, 7 figures)(12 pages, 7 figures)(12 pages, 7 figures)We formulate and propose an algorithm (MultiRank) for the ranking of nodes and layers in large multiplex networks. MultiRank takes into account the full multiplex network structure of the data and exploits the dual nature of the network in terms of nodes and layers. The proposed centrality of the layers (influences) and the centrality of the nodes are determined by a coupled set of equations. The basic idea consists in assigning more centrality to nodes that receive links from highly influential layers and from already central nodes. The layers are more influential if highly central nodes are active in them. The algorithm applies to directed/undirected as well as to weighted/unweighted multiplex networks. We discuss the application of MultiRank to three major examples of multiplex network datasets: the European Air Transportation Multiplex Network, the Pierre Auger Multiplex Collaboration Network and the FAO Multiplex Trade Network
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
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
Multilayer Networks
In most natural and engineered systems, a set of entities interact with each
other in complicated patterns that can encompass multiple types of
relationships, change in time, and include other types of complications. Such
systems include multiple subsystems and layers of connectivity, and it is
important to take such "multilayer" features into account to try to improve our
understanding of complex systems. Consequently, it is necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community structure,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure
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