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since its origins , the new science of complex networks has been primarily driven by the need to characterize the properties of real - world systems  @xcite .
the introduction of new ideas and concepts in the field has been very often associated to the availability of new , more accurate , or larger data sets , and to the discovery of new structural properties of complex systems from the real world  @xcite .
this is the reason why a lot of interest has been recently devoted to the study of multiplex networks , i.e. networks in which the same set of nodes can be connected by means of links of qualitatively different type or nature .
several data sets of real - world systems that can be represented and studied as multiplex networks have appeared in the recent literature  @xcite , and we expect that many more will arrive in the next few years . the first papers on the subject have focused on the characterization of the structure of multiplex networks  @xcite , and on modeling the basic mechanisms of their growth  @xcite . in parallel to this , some effort has been also devoted on investigating various kinds of dynamical processes on multiplex topologies , including diffusion  @xcite , epidemic spreading  @xcite , cooperation  @xcite , opinion formation  @xcite , and percolation  @xcite .
there is today a general agreement on the fact that multiplex networks represent the ideal framework to study a large variety of complex systems of different nature .
and there are already some numerical and analytical results showing that the dynamics of processes on multiplex networks is far richer than in networks with a single layer .
a comprehensive review of the main advances in this new vibrant field of research can be found in a few recent survey papers  @xcite .    in this article
we focus on an issue that has revealed of great importance in single - layer networks , but has not yet been investigated thoroughly in multiplex networks , i.e. that of correlations  @xcite . in networks with a single layer it has been found that there are correlations in the properties of connected nodes .
namely , the degree of a node can be either positively or negatively correlated with the degree of its first neighbors . in the first case , the hubs of the networks
are preferentially linked to each others , while in the second case they are preferentially connected to low - degree nodes  @xcite .    in multiplex networks
the very same concept of correlations is far richer than in a network with a single layer .
in fact , on one hand it is still possible to explore the standard degree - degree correlations at the level of each layer of the network , but on the other hand it is more interesting to introduce a truly multiplex definition of correlations , for instance by looking at how a certain property of a node at a given layer is correlated to the same or other properties of the same node at another layer .
we present here a complete and self - consistent study of correlations of node properties in multiplex networks . in doing this , we follow the usual steps of the typical approach to complex networks : _
i ) _ we first explore empirically correlations in real multiplex networks , _ ii ) _ we introduce various measures to characterize and quantify correlations in multiplex networks , _ iii ) _ we propose a series of models to reproduce the correlations found in real multiplex systems , or to assess their relevance .
we find that multiplexity introduces novel levels of complexity .
in particular , in real - world multiplex networks the patterns of presence and involvement of the nodes at the different layers are characterized by strong correlations . and
this has to be taken into account when it comes to model such systems .
the article is organized as follows . in sect .
[ why ] we focus on two small real - world networks and we use them as examples to explain why a description in terms of multiplex networks captures more information on a system than a single - layer projection . in the remaining sections we study the structure of five real - world multiplex networks ( additional information about the networks are provided in appendix ) , with the main attention to the concept of correlations . in particular , in sect .
[ sec : activity ] we focus on the patterns of node activity and involvement at the various layers .
we say that a node is active at a given layer if it has at least one link at that layer , and we introduce various quantities to characterize the distribution and the correlations of node activities .
we also investigate the activity correlations between pairs of layers .
we find that real - world multiplex networks are quite sparse , with only a few nodes active in many layers , and are characterized by strong correlations : interestingly , the activity of a node in a particular layer is very often correlated with its activity in some other layer . in sect .
[ sec : activity_models ] we introduce the first null - models to assess the significance of the observed patterns of node activity . in sect .
[ sec : cartography ] we investigate correlations between the activity and the degree of the nodes of a multiplex network , while in sect .
[ sec : degree - degree ] we show how to quantify inter - layer degree correlations ( degree correlations between layers ) .
in particular , we focus our attention on measuring correlation in the node degrees of a pair of layers , either by using the spearman s rank correlation coefficient of the two degree sequences , or by plotting , as a function of @xmath0 , the average degree @xmath1 at one layer of nodes having degree @xmath0 at the other layer .
we find that there exist significant correlations among the degree of the same node at different layers , and such correlations can be either positive , meaning that nodes tend to have similar roles across layers , or negative , meaning that nodes with a large degree in one layer tend to have small degrees in another layer . finally , in sect.[sec : model_degree - degree
] , we propose two algorithms based on simulated annealing which allow to construct multiplex networks with tunable inter - layer degree - degree correlations , and in sect .  [
sec : conclusions ] we report our conclusions .
the details on the five multiplex networks constructed from data sets of biological , technological and social complex systems , and analyzed in the paper , are describe in the appendix .
the networks and the software implementations of the algorithms described in this paper are available at  @xcite .
the aim of this work is to identify , measure and model the different kinds of correlations among node properties which can be found in a multiplex network .
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the interactions among the elementary components of many complex systems can be qualitatively different . 
 such systems are therefore naturally described in terms of multiplex or multi - layer networks , i.e.  networks where each layer stands for a different type of interaction between the same set of nodes . 
 there is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single - layer projection . here , we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks . 
 correlations in node properties , especially degree - degree correlations , have been thoroughly studied in single - layer networks . 
 here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network . 
 such correlations are intrinsically multiplex , and we first study them empirically by constructing and analyzing several multiplex networks from the real - world . in particular , we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers , and between activities and degrees . 
 we show that real - world networks exhibit indeed non - trivial multiplex correlations . 
 for instance , we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees , while other two layers are negatively correlated . 
 we then focus on constructing synthetic multiplex networks , proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance .