Many recent developments in network analysis have focused on multilayer networks, which one can use to encode time-dependent interactions, multiple types of interactions, and other complications that arise in complex systems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features, such as community structure. A prominent type of method for inferring such structures is the employment of multilayer stochastic block models (SBMs). A common (but {potentially} inadequate) assumption of these models is the sampling of edges in different layers independently, conditioned on the community labels of the nodes. In this paper, we relax this assumption of independence by incorporating edge correlations into an SBM-like model. We derive maximum-likelihood estimates of the key parameters of our model, and we propose a measure of layer correlation that reflects the similarity between connectivity patterns in different layers. Finally, we explain how to use correlated models for edge "prediction" (i.e., inference) in multilayer networks. By taking into account edge correlations, prediction accuracy improves both in synthetic networks and in a temporal network of shoppers who are connected to previously-purchased grocery products.
A wide range of complex systems can be modeled as networks with corresponding constraints on the edges and nodes, which have been extensively studied in recent years. Nowadays, with the progress of information technology, systems that contain the information collected from multiple perspectives have been generated. The conventional models designed for single perspective networks fail to depict the diverse topological properties of such systems, so multilayer network models aiming at describing the structure of these networks emerge. As a major concern in network science, decomposing the networ…
Here we introduce probabilistic weighted and unweighted multilayer networks as derived from information theoretical correlation measures on large multidimensional datasets. We present the fundamentals of the formal application of probabilistic inference on problems embedded in multilayered environments, providing examples taken from the analysis of biological and social systems: cancer genomics and drug-related violence.
We study diffusion on a multilayer network where the contact dynamics between the nodes is governed by a random process and where the waiting time distribution differs for edges from different layers. We study the impact on a random walk of the competition that naturally emerges between the edges of the different layers. In opposition to previous studies which have imposed a priori inter-layer competition, the competition is here induced by the heterogeneity of the activity on the different layers. We first study the precedence relation between different edges and by extension between differen…
Multilayer networks are a useful data structure for simultaneously capturing multiple types of relationships between a set of nodes. In such networks, each relational definition gives rise to a layer. While each layer provides its own set of information, community structure across layers can be collectively utilized to discover and quantify underlying relational patterns between nodes. To concisely extract information from a multilayer network, we propose to identify and combine sets of layers with meaningful similarities in community structure. In this paper, we describe the "strata multilaye…
Networks are a convenient way to represent complex systems of interacting entities. Many networks contain "communities" of nodes that are more densely connected to each other than to nodes in the rest of the network. In this paper, we investigate the detection of communities in temporal networks represented as multilayer networks. As a focal example, we study time-dependent financial-asset correlation networks. We first argue that the use of the "modularity" quality function---which is defined by comparing edge weights in an observed network to expected edge weights in a "null network"---is ap…
Multilayer networks capture pairwise relationships between the components of complex systems across multiple modes or scales of interactions. An important meso-scale feature of these networks is measured though their community structure, which defines groups of strongly connected nodes that exist within and across network layers. Because interlayer edges can describe relationships between different modalities, scales, or time points, it is essential to understand how communities change and evolve across layers. A popular method for detecting communities in multilayer networks consists of maxim…
Network representations are useful for describing the structure of a large variety of complex systems. Although most studies of real-world networks suppose that nodes are connected by only a single type of edge, most natural and engineered systems include multiple subsystems and layers of connectivity. This new paradigm has attracted a great deal of attention and one fundamental challenge is to characterize multilayer networks both structurally and dynamically. One way to address this question is to study the spectral properties of such networks. Here, we apply the framework of graph quotients…
The analysis of multilayer networks is among the most active areas of network science, and there are now several methods to detect dense "communities" of nodes in multilayer networks. One way to define a community is as a set of nodes that trap a diffusion-like dynamical process (usually a random walk) for a long time. In this view, communities are sets of nodes that create bottlenecks to the spreading of a dynamical process on a network. We analyze the local behavior of different random walks on multiplex networks (which are multilayer networks in which different layers correspond to differen…
We introduce a novel model for multilayer weighted networks that accounts for global noise in addition to local signals. The model is similar to a multilayer stochastic blockmodel (SBM), but the key difference is that between-block interactions independent across layers are common for the whole system, which we call ambient noise. A single block is also characterized by these fixed ambient parameters to represent members that do not belong anywhere else. This approach allows simultaneous clustering and typologizing of blocks into signal or noise in order to better understand their roles in the…
We are interested in the widespread problem of clustering documents and finding topics in large collections of written documents in the presence of metadata and hyperlinks. To tackle the challenge of accounting for these different types of datasets, we propose a novel framework based on Multilayer Networks and Stochastic Block Models. The main innovation of our approach over other techniques is that it applies the same non-parametric probabilistic framework to the different sources of datasets simultaneously. The key difference to other multilayer complex networks is the strong unbalance betwe…
Community detection in networks is commonly performed using information about interactions between nodes. Recent advances have been made to incorporate multiple types of interactions, thus generalizing standard methods to multilayer networks. Often though, one can access additional information regarding individual nodes, attributes or covariates. A relevant question is thus how to properly incorporate this extra information in such frameworks. Here we develop a method that incorporates both the topology of interactions and node attributes to extract communities in multilayer networks. We propo…
The University of Western Ontario（西安大略大学）；Reykjavik University（雷克雅未克大学）
In this paper we present a novel algorithm to study the evolution of credit risk across complex multilayer networks. Pagerank-like algorithms allow for the propagation of an influence variable across single networks, and allow quantifying the risk single entities (nodes) are subject to given the connection they have to other nodes in the network. Multilayer networks, on the other hand, are networks where subset of nodes can be associated to a unique set (layer), and where edges connect elements either intra or inter networks. Our personalized PageRank algorithm for multilayer networks allows f…
In this work, the aim is to study the spread of a contagious disease and information on a multilayer social system. The main idea is to find a criterion under which the adoption of the spreading information blocks or suppresses the epidemic spread. A two-layer network is the base of the model. The first layer describes the direct contact interactions, while the second layer is the information propagation layer. Both layers consist of the same nodes. The society consists of five different categories of individuals: susceptibles, infective, recovered, vaccinated and precautioned. Initially, only…
Characterizing large-scale organization in networks, including multilayer networks, is one of the most prominent topics in network science and is important for many applications. One type of mesoscale feature is community structure, in which sets of nodes are densely connected internally but sparsely connected to other dense sets of nodes. Two of the most popular approaches for community detection are to maximize an objective function called "modularity" and to perform statistical inference using stochastic block models. Generalizing work by Newman on monolayer networks (Physical Review E 94, …
Many systems are naturally represented by a multilayer network in which edges exist in multiple layers that encode different, but potentially related, types of interactions, and it is important to understand limitations on the detectability of community structure in these networks. Using random matrix theory, we analyze detectability limitations for multilayer (specifically, multiplex) stochastic block models (SBMs) in which L layers are derived from a common SBM. We study the effect of layer aggregation on detectability for several aggregation methods, including summation of the layers' adjac…
Multilayer networks have been found to be prone to abrupt cascading failures under random and targeted attacks, but most of the targeting algorithms proposed so far have been mainly tested on uncorrelated systems. Here we show that the size of the critical percolation set of a multilayer network is substantially affected by the presence of inter-layer degree correlations and edge overlap. We provide extensive numerical evidence which confirms that the state-of-the-art optimal percolation strategies consistently fail to identify minimal percolation sets in synthetic and real-world correlated mu…
We address the issue of the reducibility of the dynamics on a multilayer network to an equivalent process on an aggregated single-layer network. As a typical example of models for opinion formation in social networks, we implement the voter model on a two-layer multiplex network, and we study its dynamics as a function of two control parameters, namely the fraction of edges simultaneously existing in both layers of the network (edge overlap), and the fraction of nodes participating in both layers (interlayer connectivity or degree of multiplexity). We compute the asymptotic value of the number…
Complex systems are often characterized by distinct types of interactions between the same entities. These can be described as a multilayer network where each layer represents one type of interaction. These layers may be interdependent in complicated ways, revealing different kinds of structure in the network. In this work we present a generative model, and an efficient expectation-maximization algorithm, which allows us to perform inference tasks such as community detection and link prediction in this setting. Our model assumes overlapping communities that are common between the layers, while…
University of Exeter（埃克塞特大学）；University of Pisa（比萨大学）；Institute of Information Science and Technologies, National Research Council（国家研究委员会信息科学与技术研究所）；University of Ghent（根特大学）；Technion, Israel Institute of Technology（以色列理工学院）；University of Kansas（堪萨斯大学）
The mental lexicon is a complex cognitive system representing information about the words/concepts that one knows. Decades of psychological experiments have shown that conceptual associations across multiple, interactive cognitive levels can greatly influence word acquisition, storage, and processing. How can semantic, phonological, syntactic, and other types of conceptual associations be mapped within a coherent mathematical framework to study how the mental lexicon works? We here review cognitive multilayer networks as a promising quantitative and interpretative framework for investigating t…
Network analysis has played a key role in knowledge discovery and data mining. In many real-world applications in recent years, we are interested in mining multilayer networks, where we have a number of edge sets called layers, which encode different types of connections and/or time-dependent connections over the same set of vertices. Among many network analysis techniques, dense subgraph discovery, aiming to find a dense component in a network, is an essential primitive with a variety of applications in diverse domains. In this paper, we introduce a novel optimization model for dense subgraph…
Networks extracted from social media platforms frequently include multiple types of links that dynamically change over time; these links can be used to represent dyadic interactions such as economic transactions, communications, and shared activities. Organizing this data into a dynamic multiplex network, where each layer is composed of a single edge type linking the same underlying vertices, can reveal interesting cross-layer interaction patterns. In coevolving networks, links in one layer result in an increased probability of other types of links forming between the same node pair. Hence we …
Mobile phone data have recently become an attractive source of information about mobility behavior. Since cell phone data can be captured in a passive way for a large user population, they can be harnessed to collect well-sampled mobility information. In this paper, we propose CT-Mapper, an unsupervised algorithm that enables the mapping of mobile phone traces over a multimodal transport network. One of the main strengths of CT-Mapper is its capability to map noisy sparse cellular multimodal trajectories over a multilayer transportation network where the layers have different physical properti…
Networks (or graphs) are used to model the dyadic relations between entities in a complex system. In cases where there exists multiple relations between the entities, the complex system can be represented as a multilayer network, where the network in each layer represents one particular relation (or feature). The analysis of multilayer networks involves combining edges from specific layers and then computing a network property. Different subsets of the layers can be combined. For any Boolean combination operation (e.g. AND, OR), the number of possible subsets is exponential to the number of la…
