The controllability of a network is a theoretical problem of relevance in a variety of contexts ranging from financial markets to the brain. Until now, network controllability has been characterized only on isolated networks, while the vast majority of complex systems are formed by multilayer networks. Here we build a theoretical framework for the linear controllability of multilayer networks by mapping the problem into a combinatorial matching problem. We found that correlating the external signals in the different layers can significantly reduce the multiplex network robustness to node removal, as it can be seen in conjunction with a hybrid phase transition occurring in interacting Poisson networks. Moreover we observe that multilayer networks can stabilize the fully controllable multiplex network configuration that can be stable also when the full controllability of the single network is not stable.
The study of how diseases spread has greatly benefited from advances in network modeling. Recently, a class of networks known as multilayer graphs has been shown to describe more accurately many real systems, making it possible to address more complex scenarios in epidemiology such as the interaction between different pathogens or multiple strains of the same disease. In this work, we study in depth a class of networks that have gone unnoticed up to now, despite of its relevance for spreading dynamics. Specifically, we focus on directed multilayer networks, characterized by the existence of di…
We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterog…
Multilayer networks represent systems in which there are several topological levels each one representing one kind of interaction or interdependency between the systems' elements. These networks have attracted a lot of attention recently because their study allows considering different dynamical modes concurrently. Here, we revise the main concepts and tools developed up to date. Specifically, we focus on several metrics for multilayer network characterization as well as on the spectral properties of the system, which ultimately enable for the dynamical characterization of several critical phe…
Multilayer network analysis is a useful approach for studying the structural properties of entities with diverse, multitudinous relations. Classifying the importance of nodes and node-layer tuples is an important aspect of the study of multilayer networks. To do this, it is common to calculate various centrality measures, which allow one to rank nodes and node-layers according to a variety of structural features. In this paper, we formulate occupation, PageRank, betweenness, and closeness centralities in terms of node-occupation properties of different types of continuous-time classical and qu…
The Susceptible-Infected-Recovered (SIR) model is studied in multilayer networks with arbitrary number of links across the layers. By following the mapping to bond percolation we give the analytical expression for the epidemic threshold and the fraction of the infected individuals in arbitrary number of layers. These results provide an exact prediction of the epidemic threshold for infinite locally tree-like multilayer networks, and an lower bound of the epidemic threshold for more general multilayer networks. The case of a multilayer network formed by two interconnected networks is specifical…
Network motifs can capture basic interaction patterns and inform the functional properties of networks. However, real-world complex systems often have multiple types of relationships, which cannot be represented by a monolayer network. The multilayer nature of complex systems demands research on extending the notion of motifs to multilayer networks, thereby exploring the interaction patterns with a higher resolution. In this paper, we propose a formal definition of multilayer motifs, and analyse the occurrence of three-node multilayer motifs in a set of real-world multilayer networks. We find …
The field of neuroscience is facing an unprecedented expanse in the volume and diversity of available data. Traditionally, network models have provided key insights into the structure and function of the brain. With the advent of big data in neuroscience, both more sophisticated models capable of characterizing the increasing complexity of the data and novel methods of quantitative analysis are needed. Recently multilayer networks, a mathematical extension of traditional networks, have gained increasing popularity in neuroscience due to their ability to capture the full information of multi-mo…
On the basis of structural model of intersystem interactions, the main local and global structural characteristics of nodes of the multilayer network (MLN) are determined. The notions of weighted and binary aggregate-networks of MLN are introduced and integral indicators of the importance of nodes of the multilayer network are determined. The effectiveness of application of aggregate-networks for solving a number of practically important problems of the theory of complex networks is shown. To determine the most important components of intersystem interactions structure, the concepts of p-core …
Representing various networked data as multiplex networks, networks of networks and other multilayer networks can reveal completely new types of structures in these system. We introduce a general and principled graphlet framework for multilayer networks which allows one to break any multilayer network into small multilayered building blocks. These multilayer graphlets can be either analyzed themselves or used to do tasks such as comparing different systems. The method is flexible in terms of multilayer isomorphism, automorphism orbit definition, and the type of multilayer network. We illustrat…
Designing an efficient routing strategy is of great importance to alleviate traffic congestion in multilayer networks. In this work, we design an effective routing strategy for multilayer networks by comprehensively considering the roles of nodes' local structures in micro-level, as well as the macro-level differences in transmission speeds between different layers. Both numerical and analytical results indicate that our proposed routing strategy can reasonably redistribute the traffic load of the low speed layer to the high speed layer, and thus the traffic capacity of multilayer networks are…
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 extend the concept of graph isomorphisms to multilayer networks with any number of "aspects" (i.e., types of layering). In developing this generalization, we identify multiple types of isomorphisms. For example, in multilayer networks with a single aspect, permuting vertex labels, layer labels, and both vertex labels and layer labels each yield different isomorphism relations between multilayer networks. Multilayer network isomorphisms lead naturally to defining isomorphisms in any of the numerous types of networks that can be represented as a multilayer network, and we thereby obtain isomo…
Faculty of Maritime Studies and Transport, University of Ljubljana（卢布尔雅那大学海事研究与运输学院）；Department of Electronics-ICT, Faculty of Applied Engineering, University of Antwerp（安特卫普大学应用工程学院电子与信息技术系）；Department of Transport & Planning, Delft University of Technology（代尔夫特理工大学交通与规划系）
We study localization properties of principal eigenvector (PEV) of multilayer networks. Starting with a multilayer network corresponding to a delocalized PEV, we rewire the network edges using an optimization technique such that the PEV of the rewired multilayer network becomes more localized. The framework allows us to scrutinize structural and spectral properties of the networks at various localization points during the rewiring process. We show that rewiring only one-layer is enough to attain a multilayer network having a highly localized PEV. Our investigation reveals that a single edge re…
Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. In this paper, we numerically demonstrate that the normalized localization length $β$ of the eigenfunctions of multilayer random networks follows a simple scaling law given by $β=x^*/(1+x^*)$, with $x^*=γ(b_{\text{eff}}^2/L)^δ$, $γ,δ\sim 1$ and $b_{\text{eff}}$ being the effective bandwidth of the adjacency matrix of the network, whose size is $L=M\times N$. …
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 fashio…
Complex systems are characterized by many interacting units that give rise to emergent behavior. A particularly advantageous way to study these systems is through the analysis of the networks that encode the interactions among the system's constituents. During the last two decades, network science has provided many insights in natural, social, biological and technological systems. However, real systems are more often than not interconnected, with many interdependencies that are not properly captured by single layer networks. To account for this source of complexity, a more general framework, i…
Multilayer networks are a useful way to capture and model multiple, binary or weighted relationships among a fixed group of objects. While community detection has proven to be a useful exploratory technique for the analysis of single-layer networks, the development of community detection methods for multilayer networks is still in its infancy. We propose and investigate a procedure, called Multilayer Extraction, that identifies densely connected vertex-layer sets in multilayer networks. Multilayer Extraction makes use of a significance based score that quantifies the connectivity of an observe…
The approach to decomposition of multidimensional network systems on monoflow multilayer network systems (MLNS) is proposed, which significantly simplifies the analysis of intersystem interactions in multipurpose and multifunctional supersystem formations. A flow model of monoflow partially overlapped MLNS is built, which reflects the process of intersystem interactions between network systems that ensure the movement of flows of a certain type. The main local and global flow characteristics of MLNS elements are determined, which allow us more adequately from a functional point of view to esta…
A susceptible-infected-susceptible (SIS) model of multiple contagions on multilayer networks is developed to incorporate different spreading channels and disease mutations. The basic reproduction number for this model is estimated analytically. In a special case when considering only compartmental models, we analytically analyze an example of a model with a mutation driven strain persistence characterized by the absence of an epidemic threshold. This model is not related to the network topology and can be observed in both compartmental models and models on networks. The novel multiple-contagio…
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 aggr…
Information spreading is an interesting field in the domain of online social media. In this work, we are investigating how well different seed selection strategies affect the spreading processes simulated using independent cascade model on eighteen multilayer social networks. Fifteen networks are built based on the user interaction data extracted from Facebook public pages and tree of them are multilayer networks downloaded from public repository (two of them being Twitter networks). The results indicate that various state of the art seed selection strategies for single-layer networks like K-S…
Multilayer networks allow one to represent diverse and coupled connectivity patterns --- e.g., time-dependence, multiple subsystems, or both --- that arise in many applications and which are difficult or awkward to incorporate into standard network representations. In the study of multilayer networks, it is important to investigate mesoscale (i.e., intermediate-scale) structures, such as dense sets of nodes known as communities, to discover network features that are not apparent at the microscale or the macroscale. The ill-defined nature of mesoscale structure and its ubiquity in empirical net…
Several systems can be modeled as sets of interconnected networks or networks with multiple types of connections, here generally called multilayer networks. Spreading processes such as information propagation among users of an online social networks, or the diffusion of pathogens among individuals through their contact network, are fundamental phenomena occurring in these networks. However, while information diffusion in single networks has received considerable attention from various disciplines for over a decade, spreading processes in multilayer networks is still a young research area prese…
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.
Complex systems are usually illustrated by networks which captures the topology of the interactions between the entities. To better understand the roles played by the entities in the system one needs to uncover the underlying community structure of the system. In recent years, systems with interactions that have various types or can change over time between the entities have attracted an increasing research attention. However, algorithms aiming to solve the key problem - community detection - in multilayer networks are still limited. In this work, we first introduce the multilayer network mode…
Multilayer networks have been widely used to represent and analyze systems of interconnected entities where both the entities and their connections can be of different types. However, real multilayer networks can be difficult to analyze because of irrelevant information, such as layers not related to the objective of the analysis, because of their size, or because traditional methods defined to analyze simple networks do not have a straightforward extension able to handle multiple layers. Therefore, a number of methods have been devised in the literature to simplify multilayer networks with th…
This paper aims to establish theoretical foundations of graph product multilayer networks (GPMNs), a family of multilayer networks that can be obtained as a graph product of two or more factor networks. Cartesian, direct (tensor), and strong product operators are considered, and then generalized. We first describe mathematical relationships between GPMNs and their factor networks regarding their degree/strength, adjacency, and Laplacian spectra, and then show that those relationships can still hold for nonsimple and generalized GPMNs. Applications of GPMNs are discussed in three areas: predict…
The problem of node-centric, or local, community detection in information networks refers to the identification of a community for a given input node, having limited information about the network topology. Existing methods for solving this problem, however, are not conceived to work on complex networks. In this paper, we propose a novel framework for local community detection based on the multilayer network model. Our approach relies on the maximization of the ratio between the community internal connection density and the external connection density, according to multilayer similarity-based c…
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…
Presented data contains the record of five spreading campaigns that occurred in a virtual world platform. Users distributed avatars between each other during the campaigns. The processes varied in time and range and were either incentivized or not incentivized. Campaign data is accompanied by events. The data can be used to build a multilayer network to place the campaigns in a wider context. To the best of the authors knowledge, the study is the first publicly available dataset containing a complete real multilayer social network together, along with five complete spreading processes in it.
Multilayer network analysis has become a vital tool for understanding different relationships and their interactions in a complex system, where each layer in a multilayer network depicts the topological structure of a group of nodes corresponding to a particular relationship. The interactions among different layers imply how the interplay of different relations on the topology of each layer. For a single-layer network, network embedding methods have been proposed to project the nodes in a network into a continuous vector space with a relatively small number of dimensions, where the space embed…
This paper provides a multilayer and time-varying structural analysis of one air transportation network, having the Brazilian air transportation network as a case study. Using a single mathematical object called MultiAspect Graph (MAG) for this analysis, the multilayer perspective enables the unveiling of the particular strategies of each airline to both establish and adapt in a moment of crisis its specific flight network. Similarly, the time-varying perspective allows multi-scale analysis considering different time periods, and thus assessing the impact of the economic crisis on how the diff…
The use of patent citation networks as research tools is becoming increasingly commonplace in the field of innovation studies. However, these networks rarely consider the contexts in which these citations are generated and are generally restricted to a single jurisdiction. Here, we propose and explore the use of a multilayer network framework that can naturally incorporate citation metadata and stretch across jurisdictions, allowing for a complete view of the global technological landscape that is accessible through patent data. Taking a conservative approach that links citation network layers…
Multilayer networks allow for modeling complex relationships, where individuals are embedded in multiple social networks at the same time. Given the ubiquity of such relationships, these networks have been increasingly gaining attention in the literature. This paper presents the first analysis of the robustness of centrality measures against strategic manipulation in multilayer networks. More specifically, we consider an "evader" who strategically chooses which connections to form in a multilayer network in order to obtain a low centrality-based ranking-thereby reducing the chance of being hig…
The generalized version of modularity for multilayer networks, a.k.a. multislice modularity, is characterized by two model parameters, namely resolution factor and inter-layer coupling factor. The former corresponds to a notion of layer-specific relevance, whereas the inter-layer coupling factor represents the strength of node connections across the network layers. Despite the potential of this approach, the setting of both parameters can be arbitrarily selected, without considering specific characteristics from the topology of the multilayer network as well as from an available community stru…
In this paper, we present the model of the interaction between the spread of disease and the spread of information about the disease in multilayer networks. Next, based on the characteristics of the SARS-COV-2 virus pandemic, we evaluated the influence of information blocking on the virus spread. Our results show that blocking the spread of information affects the speed at which the epidemic peak appears in our society and the number of infected individuals.
Modularity for multilayer networks, also called multislice modularity, is parametric to a resolution factor and an inter-layer coupling factor. The former is useful to express layer-specific relevance and the latter quantifies the strength of node linkage across the layers of a network. However, such parameters can be set arbitrarily, thus discarding any structure information at graph or community level. Other issues are related to the inability of properly modeling order relations over the layers, which is required for dynamic networks. In this paper we propose a new definition of modularity …
To improve the accuracy of network-based SIS models we introduce and study a multilayer representation of a time-dependent network. In particular, we assume that individuals have their long-term (permanent) contacts that are always present, identifying in this way the first network layer. A second network layer also exists, where the same set of nodes can be connected by occasional links, created with a given probability. While links of the first layer are permanent, a link of the second layer is only activated with some probability and under the condition that the two nodes, connected by this…
Various company interactions can be described by networks, for instance the ownership networks and the board membership networks. To understand the ecosystem of companies, these interactions cannot be seen in isolation. For this purpose we construct a new multilayer network of interactions between companies in Germany and in the United Kingdom, combining ownership links, social ties through joint board directors, R\&D collaborations and stock correlations in one linked multiplex dataset. We describe the features of this network and show there exists a non-trivial overlap between these differen…
The paper introduces a number of structural and functional features of classification of multilayer networks (MLN), by means of which distinguish monoflow partially overlapped MLN, that are quite common in the study of intersystem interactions of different nature. The concept of MLN's aggregate-network is defined, which in many cases significantly simplifies the study of intersystem interactions, and the properties of its k-cores are investigated. The notion of p-cores is introduced, with help of which the components of MLN that are directly involved in the implementation of intersystem intera…
Multilayer relationships among entities and information about entities must be accompanied by the means to analyze, visualize, and obtain insights from such data. We present open-source software (muxViz) that contains a collection of algorithms for the analysis of multilayer networks, which are an important way to represent a large variety of complex systems throughout science and engineering. We demonstrate the ability of muxViz to analyze and interactively visualize multilayer data using empirical genetic, neuronal, and transportation networks. Our software is available at https://github.com…
Population analyses of functional connectivity have provided a rich understanding of how brain function differs across time, individual, and cognitive task. An important but challenging task in such population analyses is the identification of reliable features that describe the function of the brain, while accounting for individual heterogeneity. Our work is motivated by two particularly important challenges in this area: first, how can one analyze functional connectivity data over populations of individuals, and second, how can one use these analyses to infer group similarities and differenc…
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…
Centrality is widely recognized as one of the most critical measures to provide insight in the structure and function of complex networks. While various centrality measures have been proposed for single-layer networks, a general framework for studying centrality in multilayer networks (i.e., multicentrality) is still lacking. In this study, a tensor-based framework is introduced to study eigenvector multicentrality, which enables the quantification of the impact of interlayer influence on multicentrality, providing a systematic way to describe how multicentrality propagates across different la…
Unveiling the community structure of networks is a powerful methodology to comprehend interconnected systems across the social and natural sciences. To identify different types of functional modules in interaction data aggregated in a single network layer, researchers have developed many powerful methods. For example, flow-based methods have proven useful for identifying modular dynamics in weighted and directed networks that capture constraints on flow in the systems they represent. However, many networked systems consist of agents or components that exhibit multiple layers of interactions. I…
Multilayer networks preserve full information about the different interactions among the constituents of a complex system, and have recently proven quite useful in modelling transportation networks, social circles, and the human brain. A fundamental and still open problem is to assess if and when the multilayer representation of a system provides a qualitatively better model than the classical single-layer aggregated network. Here we tackle this problem from an algorithmic information theory perspective. We propose an intuitive way to encode a multilayer network into a bit string, and we defin…
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…
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…
Recently, Social Network Analysis studies have led to an improvement and to a generalization of existing tools to networks with multiple subsystems and layers of connectivity. These kind of networks are usually called multilayer networks. Multilayer networks in which each layer shares at least one node with some other layer in the network are called multiplex networks. Being a multiplex network does not require all nodes to exist on every layer. In this paper, we built a criminal multiplex network which concerns an anti-mafia operation called "Montagna" and it is based on the examination of a …
Online social systems are multiplex in nature as multiple links may exist between the same two users across different social networks. In this work, we introduce a framework for studying links and interactions between users beyond the individual social network. Exploring the cross-section of two popular online platforms - Twitter and location-based social network Foursquare - we represent the two together as a composite multilayer online social network. Through this paradigm we study the interactions of pairs of users differentiating between those with links on one or both networks. We find th…
The amount and variety of data is increasing drastically for several years. These data are often represented as networks, which are then explored with approaches arising from network theory. Recent years have witnessed the extension of network exploration methods to leverage more complex and richer network frameworks. Random walks, for instance, have been extended to explore multilayer networks. However, current random walk approaches are limited in the combination and heterogeneity of network layers they can handle. New analytical and numerical random walk methods are needed to cope with the …
