In complex systems, the network of interactions we observe between system's components is the aggregate of the interactions that occur through different mechanisms or layers. Recent studies reveal that the existence of multiple interaction layers can have a dramatic impact in the dynamical processes occurring on these systems. However, these studies assume that the interactions between systems components in each one of the layers are known, while typically for real-world systems we do not have that information. Here, we address the issue of uncovering the different interaction layers from aggregate data by introducing multilayer stochastic block models (SBMs), a generalization of single-layer SBMs that considers different mechanisms of layer aggregation. First, we find the complete probabilistic solution to the problem of finding the optimal multilayer SBM for a given aggregate observed network. Because this solution is computationally intractable, we propose an approximation that enables us to verify that multilayer SBMs are more predictive of network structure in real-world complex systems. The development of tools for the analysis of real-world complex networks has significantly advanced our understanding of complex systems in fields as diverse as molecular and cell biology [1], neuroscience [2], biomedicine [3, 4] , ecology [5, 6] , economics [7] , and sociology [8] . One of the main successes of the network approach has been to unravel the relationship between the modular organization of interactions within a complex system [9] , and the function and temporal evolution of the system [10-13]. As a result, a large body of research has been devoted to the detection of the modular structure (or community structure) of complex networks, that is, to the division of the nodes of the network into densely connected subgroups [14] . Stochastic block models (SBMs) [15] [16] [17] are a class of probabilistic generative network models that provide a more general description of the (mesoscopic) group structure of real-world networks than modular models. In SBMs, nodes are assumed to belong to groups and connect to each other with probabilities that depend only on their group memberships. The simple mathematical form of SBMs has enabled not only the identification of generalized community structures in networks [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] , but also to make network inference a predictive tool to detect missing and spurious links in empirical network data [27] , to predict human decisions [28, 29] and the appearance of conflict in work teams [30] , and for the identification of unknown interactions between drugs [31]. While these approaches have pushed forward our understanding of complex network structure, a limitation is that they rely on the premise that there is a single mechanism that describes the connectivity of the network, even though we know that real-world networks are often the result of processes occurring on different "layers" (for example, social networks comprise relationships that arise in the familiar layer, and others that arise in the professional layer) [32] . Moreover, it is increasingly clear that the multilayer structure of complex networks can have a dramatic impact on the dynamical processes * Corresponding author: roger.guimera@urv.cat † Corresponding author: marta.sales@urv.cat that take place on them [33] [34] [35] [36] [37] . Unfortunately, we often lack information about the different layers of interaction and can only observe projections of these multilayer interactions into an aggregate network in which all links are equivalent. Here, we precisely address the problem of unraveling the underlying multilayer structure in real-world networks. To do so, we first introduce the family of multilayer SBMs that generalizes single-layer SBMs to situations where links arise in different layers and are aggregated through different mechanisms. Although there have been proposals to extend the concept of modularity to multilayer networks [38] , ours represents a pioneering attempt to extend generative group-based models to multilayer systems, and to study those models rigorously using tools from statistical physics. Second, we give the probabilistically complete solution to the problem of inferring the optimal multilayer SBM for a given aggregate network. Because this solution is computationally intractable, we propose an approximation which enables us to objectively address the question of whether an observed network is likely to be the projection of multiple layers. Our results suggest that many real-world networks are indeed projections.
In Ref. REF , the authors defined a version of multilayer stochastic block model and an inference procedure for assessing whether or not a single-layer network was actually obtained from an aggregation of layers in a multilayer network; they considered the aggregation of layers using boolean rules.
Multilayer stochastic block models reveal the multilayer structure of complex networks
Phys. Rev. X 6, 011036 (2016)
Physics Computer Science