Real-world Multiplex Networks Research Articles

Reconstruction of multiplex networks via graph embeddings.

Multiplex networks are collections of networks with identical nodes but distinct layers of edges. They are genuine representations of a large variety of real systems whose elements interact in multiple fashions or flavors. However, multiplex networks are not always simple to observe in the real world; often, only partial information on the layer structure of the networks is available, whereas the remaining information is in the form of aggregated, single-layer networks. Recent works have proposed solutions to the problem of reconstructing the hidden multiplexity of single-layer networks using tools proper for network science. Here, we develop a machine-learning framework that takes advantage of graph embeddings, i.e., representations of networks in geometric space. We validate the framework in systematic experiments aimed at the reconstruction of synthetic and real-world multiplex networks, providing evidence that our proposed framework not only accomplishes its intended task, but often outperforms existing reconstruction techniques.

Percolation of interlayer feature-correlated multiplex networks

Multiplex networks formed by a set of nodes connected through different types of edges are widely used to model interactions between different systems. In many real-world scenarios, the properties of nodes in different layers are characterized by non-topological features such as age, flow, and geographical location. On this basis, the associations between different layers are caused by the coupling of features. Here, we propose a percolation framework to investigate the properties of such interlayer feature-correlated multiplex networks (FCN). Based on this framework, the robustness of the networks correlated through completely unequal discrete features, discrete features with repetition, and continuous features is analyzed. Theoretical and numerical results show that the interlayer degree–degree correlation (IDDC) of the network is controlled by both the correlation between degrees and features, and the correlation between the features of different layers. When there are repeating features, the network structure is randomized due to the decrease in the differentiation of nodes in the feature space. This randomization substantially reduces the IDDC of the network even if the above correlations are fixed. Our findings provide new insight into revealing the origins of the complexities of real-world multiplex networks.

Reconstructing Sparse Multiplex Networks with Application to Covert Networks

Network structure provides critical information for understanding the dynamic behavior of complex systems. However, the complete structure of real-world networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of networks. In this paper, we integrate the configuration model for generating random networks into an Expectation-Maximization-Aggregation (EMA) framework to reconstruct the complete structure of multiplex networks. We validate the proposed EMA framework against the Expectation-Maximization (EM) framework and random model on several real-world multiplex networks, including both covert and overt ones. It is found that the EMA framework generally achieves the best predictive accuracy compared to the EM framework and the random model. As the number of layers increases, the performance improvement of EMA over EM decreases. The inferred multiplex networks can be leveraged to inform the decision-making on monitoring covert networks as well as allocating limited resources for collecting additional information to improve reconstruction accuracy. For law enforcement agencies, the inferred complete network structure can be used to develop more effective strategies for covert network interdiction.

Finding influential edges in multilayer networks: Perspective from multilayer diffusion model.

With the popularization of social network analysis, information diffusion models have a wide range of applications, such as viral marketing, publishing predictions, and social recommendations. The emergence of multiplex social networks has greatly enriched our daily life; meanwhile, identifying influential edges remains a significant challenge. The key problem lies that the edges of the same nodes are heterogeneous at different layers of the network. To solve this problem, we first develop a general information diffusion model based on the adjacency tensor for the multiplex network and show that the n-mode singular value can control the level of information diffusion. Then, to explain the suppression of information diffusion through edge deletion, efficient edge eigenvector centrality is proposed to identify the influence of heterogeneous edges. The numerical results from synthetic networks and real-world multiplex networks show that the proposed strategy outperforms some existing edge centrality measures. We devise an experimental strategy to demonstrate that influential heterogeneous edges can be successfully identified by considering the network layer centrality, and the deletion of top edges can significantly reduce the diffusion range of information across multiplex networks.

Application of hyperbolic geometry of multiplex networks under layer link-based attacks.

At present, network science can be considered one of the prosperous scientific fields. The multi-layered network approach is a recent development in this area and focuses on identifying the interactions of several interconnected networks. In this paper, we propose a new method for predicting redundant links for multiplex networks using the similarity criterion based on the hyperbolic distance of the node pairs. We retrieve lost links found on various attack strategies in multiplex networks by predicting redundant links in these networks using the proffered method. We applied the recommended algorithm to real-world multiplex networks, and the numerical simulations show its superiority over other advanced algorithms. During the studies and numerical simulations, the power of the hyperbolic geometry criterion over different standard and current methods based on link prediction used for network retrieval is evident, especially in the case of attacks based on the edge betweenness and random strategies illustrated in the results.

Interlayer link prediction based on multiple network structural attributes

Information may flow across multiple social network platforms and spread simultaneously on them. These platforms constitute a multiplex network in which each social network platform functions as a subnetwork. One problem in information propagation analysis on the multiplex network is to reveal the corresponding relationship of accounts belonging to the same user across different subnetworks. It can be summarized as the interlayer link prediction problem in a multiplex network. To predict more unobserved interlayer links accurately, most of the current structure-based approaches adopt an iterative strategy. However, iterative manners suffer from a big challenge of high time consumption. In this paper, we propose an interlayer link prediction framework based on multiple structural attributes (MulAtt). It calculates the matching degree of unmatched nodes once by leveraging the information of closed triad, intralayer links, matched neighbors and intralayer links of neighbors simultaneously to guarantee accuracy while reducing time consumption. We also propose a fast possible matched closed triad counting algorithm to improve the efficiency of obtaining the clues from the closed triad. Experiments on four widely used real-world multiplex networks demonstrate that the MulAtt framework can achieve better performance than several existing network structure-based methods in a non-iterative way.

Community-guided link prediction in multiplex networks

Multiplex link prediction is the problem of finding missing links between nodes based on information from other layers. Although the link prediction problem in the online social networks is studied comprehensively, most approaches only employ internal features of the under prediction layer and do not consider additional link information from other networks. Also, many existing link prediction techniques are only based on the extracted information from links or nodes. However, the information flow in many real-world systems like social networks is considered as collaborative relations on correlated groups as an alternative for individual relations. In this research, we have proposed a Community-guided Link Prediction based on External Similarity (CLPES) method for multiplex networks in which, beside nodes and links information, community information is also employed. In our proposed method, we used an evolutionary algorithm (MOEA/D-TS) for specifying the community structure of the desired network. Next, the incorporation of internal features of each layer with a new external similarity metric (ExSim) obtains the final values for the likelihood of link formation in the network. Experiments on various real-world multiplex networks prove the capability of the proposed CLPES method for producing improved results and its superiority against other link prediction algorithms.

Identifying super-spreaders in information–epidemic coevolving dynamics on multiplex networks

Identifying super-spreaders in epidemics is important to suppress the spreading of disease especially when the medical resource is limited. In the modern society, the information on epidemics transmits swiftly through various communication channels which contributes much to the suppression of epidemics. Here we study on the identification of super-spreaders in the information–disease coupled spreading dynamics. Firstly, we find that the centralities in physical contact layer are no longer effective to identify super-spreaders in epidemics, which is due to the suppression effects from the information spreading. Then by considering the structural and dynamical couplings between the communication layer and physical contact layer, we propose a centrality measure called coupling-sensitive centrality to identify super-spreaders of disease in the coevolving dynamics. Simulation results on synthesized and real-world multiplex networks show that the proposed measure is not only much more accurate than centralities on the single-layer network, but also outperforms two typical multilayer centralities in identifying super-spreaders. These findings imply that considering the structural and dynamical couplings between layers is very necessary in identifying the key roles in the coupled multilayer systems.

An information theoretic approach to link prediction in multiplex networks

The entities of real-world networks are connected via different types of connections (i.e., layers). The task of link prediction in multiplex networks is about finding missing connections based on both intra-layer and inter-layer correlations. Our observations confirm that in a wide range of real-world multiplex networks, from social to biological and technological, a positive correlation exists between connection probability in one layer and similarity in other layers. Accordingly, a similarity-based automatic general-purpose multiplex link prediction method—SimBins—is devised that quantifies the amount of connection uncertainty based on observed inter-layer correlations in a multiplex network. Moreover, SimBins enhances the prediction quality in the target layer by incorporating the effect of link overlap across layers. Applying SimBins to various datasets from diverse domains, our findings indicate that SimBins outperforms the compared methods (both baseline and state-of-the-art methods) in most instances when predicting links. Furthermore, it is discussed that SimBins imposes minor computational overhead to the base similarity measures making it a potentially fast method, suitable for large-scale multiplex networks.

Optimal percolation in correlated multilayer networks with overlap

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 multilayer networks, thus overestimating their robustness. We propose two new targeting algorithms, based on the local estimation of path disruptions away from a given node, and a family of Pareto-efficient strategies that take into account both intra-layer and inter-layer heuristics, and can be easily extended to multiplex networks with an arbitrary number of layers. We show that these strategies consistently outperform existing attacking algorithms, on both synthetic and real-world multiplex networks, and provide some interesting insights about the interplay of correlations and overlap in determining the hyperfragility of real-world multilayer networks. Overall, the results presented in the paper suggest that we are still far from having fully identified the salient ingredients determining the robustness of multiplex networks to targeted attacks.

Link prediction in real-world multiplex networks via layer reconstruction method.

Networks are invaluable tools to study real biological, social and technological complex systems in which connected elements form a purposeful phenomenon. A higher resolution image of these systems shows that the connection types do not confine to one but to a variety of types. Multiplex networks encode this complexity with a set of nodes which are connected in different layers via different types of links. A large body of research on link prediction problem is devoted to finding missing links in single-layer (simplex) networks. In recent years, the problem of link prediction in multiplex networks has gained the attention of researchers from different scientific communities. Although most of these studies suggest that prediction performance can be enhanced by using the information contained in different layers of the network, the exact source of this enhancement remains obscure. Here, it is shown that similarity w.r.t. structural features (eigenvectors) is a major source of enhancements for link prediction task in multiplex networks using the proposed layer reconstruction method and experiments on real-world multiplex networks from different disciplines. Moreover, we characterize how low values of similarity w.r.t. structural features result in cases where improving prediction performance is substantially hard.

The Feedback Vertex Set Problem of Multiplex Networks

The feedback vertex set (FVS) problem of network aims to find a vertex subset of smallest size that contains at least one vertex of each cycle of the network. The FVS problem has found wide applications, ranging from circuit design, network security and network control and observation etc. However, the majority of existing studies on FVS problem are confined to single networks. In this brief, under the consideration of one kind of control problem of multiplex network, we introduce the concept of FVS of multiplex network. The control problem of multiplex network is naturally mapped into the FVS problem of multiplex network. We then propose two simulated annealing (SA) algorithms to solve the directed FVS problem of multiplex network and undirected FVS problem of multiplex network respectively. The effectiveness of the SA algorithms is finally verified on both artificial multiplex networks and real-world multiplex networks.

K-core structure of real multiplex networks

The authors show that interlayer correlations are responsible for the strong k-core structure found in real world multiplex networks. The more heterogeneous are the degree distributions of the layers, the more pivotal is the role of degree correlations. The less heterogeneous are the degree distributions, the more crucial is the role of correlations at the level of node similarities. These findings may help in identifying influential spreaders in real world multiplex systems.

Learning embeddings for multiplex networks using triplet loss

Learning low-dimensional representations of graphs has facilitated the use of traditional machine learning techniques to solving classic network analysis tasks such as link prediction, node classification, community detection, etc. However, to date, the vast majority of these learning tasks are focused on traditional single-layer/unimodal networks and largely ignore the case of multiplex networks. A multiplex network is a suitable structure to model multi-dimensional real-world complex systems. It consists of multiple layers where each layer represents a different relationship among the network nodes. In this work, we propose MUNEM, a novel approach for learning a low-dimensional representation of a multiplex network using a triplet loss objective function. In our approach, we preserve the global structure of each layer, while at the same time fusing knowledge among different layers during the learning process. We evaluate the effectiveness of our proposed method by testing and comparing on real-world multiplex networks from different domains, such as collaboration network, protein-protein interaction network, online social network. Finally, in order to deliberately examine the effect of our model’s parameters we conduct extensive experiments on synthetic multiplex networks.

Consensus dynamics on weighted multiplex networks: a long-range interaction perspective

Real-world multiplex networks are weighted in nature, e.g. transports networks (variation in traffic loads over links), social networks (variation in communication between individuals), etc. In addition, the dynamical behavior of multiplex networks is influenced by the contribution of both local as well as global neighbors to individual network layers and the corresponding neighbors in other network layers. In the present work, we propose a method to compute the intra-layer and inter-layer edge weights of multiplex networks and study the consensus dynamics by analyzing the Laplacian spectra from the perspective of long-range interactions (LRIs) in each network layer. To this end, we extend the Kuramoto model for multiplex networks, consider synthetic multiplex networks with different topological characteristics and an empirical data-set. Our analysis reveals that in the case of LRIs, there is an improvement in the stability of the consensus process in the independent network layers as well as in the multiplex networks. However, depending upon the topology of the network layers being added to the existing multiplex network, the behavior of stability is different. In addition, we study the level of synchronization (R) by analyzing the Laplacian spectra and order parameter () of the Kuramoto model. We find that by adding more layers to the existing multiplex network, the value of R increases at the given timestamps in the case of Laplacian spectra analysis. In the case of the Kuramoto model, the order parameter shows different behavior depending upon the topological characteristics of the network layers being added to the existing multiplex network.

Application of hyperbolic geometry in link prediction of multiplex networks

Recently multilayer networks are introduced to model real systems. In these models the individuals make connection in multiple layers. Transportation networks, biological systems and social networks are some examples of multilayer networks. There are various link prediction algorithms for single-layer networks and some of them have been recently extended to multilayer networks. In this manuscript, we propose a new link prediction algorithm for multiplex networks using two novel similarity metrics based on the hyperbolic distance of node pairs. We use the proposed methods to predict spurious and missing links in multiplex networks. Missing links are those links that may appear in the future evolution of the network, while spurious links are the existing connections that are unlikely to appear if the network is evolving normally. One may interpret spurious links as abnormal links in the network. We apply the proposed algorithm on real-world multiplex networks and the numerical simulations reveal its superiority than the state-of-the-art algorithms.

Crawling the Community Structure of Multiplex Networks

We examine the problem of crawling the community structure of a multiplex network containing multiple layers of edge relationships. While there has been a great deal of work examining community structure in general, and some work on the problem of sampling a network to preserve its community structure, to the best of our knowledge, this is the first work to consider this problem on multiplex networks. We consider the specific case in which the layers of a multiplex network have different query (collection) costs and reliabilities; and a data collector is interested in identifying the community structure of the most expensive layer. We propose MultiComSample (MCS), a novel algorithm for crawling a multiplex network. MCS uses multiple levels of multi-armed bandits to determine the best layers, communities and node roles for selecting nodes to query. We test MCS against six baseline algorithms on real-world multiplex networks, and achieved large gains in performance. For example, after consuming a budget equivalent to sampling 20% of the nodes in the expensive layer, we observe that MCS outperforms the best baseline by up to 49%.

Influence of geometric correlations on epidemic spreading in multiplex networks

Real-world multiplex networks are not randomly connected by several single-layer networks. In fact, there exist significant geometric correlations among different layers. In this paper, the influence of the geometric correlations, including the radial correlation and the angular correlation, on epidemic spreading on multiplex networks is investigated. The results show that the radial correlation and/or the angular correlation can reduce the epidemic threshold and lead to smaller scale of outbreak on the artificial networks. Moreover, although geometric correlations and overlapped links are positively correlated, the influence of geometric correlations on epidemic spreading cannot be replaced by overlapped links. Simulations on a real-world multiplex network indicate that the geometric correlations can produce lower threshold and smaller scale of outbreak.

Centrality ranking in multiplex networks using topologically biased random walks

Abstract Characterizing the statistically significant centrality of nodes is one of the main goals of multiplex networks. However, current centrality measures for node rankings focus only on either random walks or on the topological structure of the network. A pressing challenge is how to measure centrality of nodes in multiplex networks, depending both on network topology and on the biased types of random walks, such as the biased walks dealing with the properties of each node separately at each layer, or the biased walks considering instead one or even more intrinsically multiplex properties of the arrival node. In the paper, considering these two aspects, we propose a mathematical framework based on topologically biased random walk, called topologically biased multiplex PageRank, which allows to calculate centrality and accordingly rank nodes in multiplex networks. In particular, depending on the nature of biases and the interaction of nodes between different layers, we distinguish additive, multiplicative and combined cases of topologically biased multiplex PageRank. Each case by tuning the bias parameters reflects how the centrality ranking of a node in one layer affects the ranking its replica can gain in the other layers, and captures the extent to which the walkers preferentially visit hubs or poorly connected nodes. Experiments on two real-world multiplex networks show that the topologically biased multiplex PageRank outperforms both its corresponding unbiased case and the current ranking methods, and it can efficiently capture the significantly top-ranked nodes in multiplex networks by means of a proper tuning of the biases in the walks.

Observability transition in multiplex networks

We extend the observability model to multiplex networks composed of two network layers. We present mathematical frameworks, valid under the treelike ansatz, able to describe the emergence of the macroscopic cluster of mutually observable nodes in both synthetic and real-world multiplex networks. We show that the observability transition in synthetic multiplex networks is discontinuous. In real-world multiplex networks instead, edge overlap among layers is responsible for the disappearance of any sign of abruptness in the emergence of the macroscopic cluster of mutually observable nodes.