Emergence of network features from multiplexity

Node-layer duality in networked systems

Real-world networks typically exhibit several aspects, or layers, of interactions among their nodes. By permuting the role of the nodes and the layers, we establish a new criterion to construct the dual of a network. This approach allows to examine connectivity from either a node-centric or layer-centric viewpoint. Through rigorous analytical methods and extensive simulations, we demonstrate that nodewise and layerwise connectivity measure different but related aspects of the same system. Leveraging node-layer duality provides complementary insights, enabling a deeper comprehension of diverse networks across social science, technology and biology. Taken together, these findings reveal previously unappreciated features of complex systems and provide a fresh tool for delving into their structure and dynamics.

Duality between predictability and reconstructability in complex systems

Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here, we find an intricate relationship between predictability and reconstructability using an information-theoretical point of view. We use the mutual information between a random graph and a stochastic process evolving on this random graph to quantify their codependence. Then, we show how the uncertainty coefficients, which are intimately related to that mutual information, quantify our ability to reconstruct a graph from an observed time series, and our ability to predict the evolution of a process from the structure of its interactions. We provide analytical calculations of the uncertainty coefficients for many different systems, including continuous deterministic systems, and describe a numerical procedure when exact calculations are intractable. Interestingly, we find that predictability and reconstructability, even though closely connected by the mutual information, can behave differently, even in a dual manner. We prove how such duality universally emerges when changing the number of steps in the process. Finally, we provide evidence that predictability-reconstruction dualities may exist in dynamical processes on real networks close to criticality.

Metrics for network comparison using egonet feature distributions

Identifying networks with similar characteristics in a given ensemble, or detecting pattern discontinuities in a temporal sequence of networks, are two examples of tasks that require an effective metric capable of quantifying network (dis)similarity. Here we propose a method based on a global portrait of graph properties built by processing local nodes features. More precisely, a set of dissimilarity measures is defined by elaborating the distributions, over the network, of a few egonet features, namely the degree, the clustering coefficient, and the egonet persistence. The method, which does not require the alignment of the two networks being compared, exploits the statistics of the three features to define one- or multi-dimensional distribution functions, which are then compared to define a distance between the networks. The effectiveness of the method is evaluated using a standard classification test, i.e., recognizing the graphs originating from the same synthetic model. Overall, the proposed distances have performances comparable to the best state-of-the-art techniques (graphlet-based methods) with similar computational requirements. Given its simplicity and flexibility, the method is proposed as a viable approach for network comparison tasks.

Analysis of Weather Factors on Aircraft Cancellation using a Multilayer Complex Network

Airlines provide one of the most popular and important transportation services for passengers. While the importance of the airline industry is rising, flight cancellations are also increasing due to abnormal weather factors, such as rainfall and wind speed. Although previous studies on cancellations due to weather factors considered both aircraft and weather factors concurrently, the complex network studies only treated the aircraft factor with a single-layer network. Therefore, the aim of this study was to apply a multilayer complex network (MCN) method that incorporated three different factors, namely, aircraft, rainfall, and wind speed, to investigate aircraft cancellations at 14 airports in the Republic of Korea. The results showed that rainfall had a greater impact on aircraft cancellations compared with wind speed. To find out the most important node in the cancellation, we applied centrality analysis based on information entropy. According to the centrality analysis, Jeju Airport was identified as the most influential node since it has a high demand for aircraft. Also, we showed that characteristics and factors of aircraft cancellation should be appropriately defined by links in the MCN. Furthermore, we verified the applicability of the MCN method in the fields of aviation and meteorology. It is expected that the suggested methodology in this study can help to understand aircraft cancellation due to weather factors.

Hierarchical core-periphery structure in networks

We study core-periphery structure in networks using inference methods based on a flexible network model that allows for traditional onionlike cores within cores, but also for hierarchical treelike structures and more general non-nested types of structures. We propose an efficient Monte Carlo scheme for fitting the model to observed networks and report results for a selection of real-world data sets. Among other things, we observe an empirical distinction between networks showing traditional core-periphery structure with a dense core weakly connected to a sparse periphery, and an alternative structure in which the core is strongly connected both within itself and to the periphery. Networks vary in whether they are better represented by one type of structure or the other. We also observe structures that are a hybrid between core-periphery structure and community structure, in which networks have a set of nonoverlapping cores that roughly correspond to communities, surrounded by a single undifferentiated periphery. Computer code implementing our methods is available.

Synchronization Induced by Layer Mismatch in Multiplex Networks

Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in terms of parameter mismatch among the layers, has not been adequately investigated. Here, we study the intralayer synchrony in multiplex networks, where the layers are different from one other, due to parameter mismatch in their local dynamics. In such a multiplex network, the intralayer coupling strength for the emergence of intralayer synchronization decreases upon the introduction of impurity among the layers, which is caused by a parameter mismatch in their local dynamics. Furthermore, the area of occurrence of intralayer synchronization also widens with increasing mismatch. We analytically derive a condition under which the intralayer synchronous solution exists, and we even sustain its stability. We also prove that, in spite of the mismatch among the layers, all the layers of the multiplex network synchronize simultaneously. Our results indicate that a multiplex network with mismatched layers can induce synchrony more easily than a multiplex network with identical layers.

Local Phase Transitions in a Model of Multiplex Networks with Heterogeneous Degrees and Inter-Layer Coupling

Multilayer networks represent multiple types of connections between the same set of nodes. Clearly, a multilayer description of a system adds value only if the multiplex does not merely consist of independent layers. In real-world multiplexes, it is expected that the observed inter-layer overlap may result partly from spurious correlations arising from the heterogeneity of nodes, and partly from true inter-layer dependencies. It is therefore important to consider rigorous ways to disentangle these two effects. In this paper, we introduce an unbiased maximum entropy model of multiplexes with controllable intra-layer node degrees and controllable inter-layer overlap. The model can be mapped to a generalized Ising model, where the combination of node heterogeneity and inter-layer coupling leads to the possibility of local phase transitions. In particular, we find that node heterogeneity favors the splitting of critical points characterizing different pairs of nodes, leading to link-specific phase transitions that may, in turn, increase the overlap. By quantifying how the overlap can be increased by increasing either the intra-layer node heterogeneity (spurious correlation) or the strength of the inter-layer coupling (true correlation), the model allows us to disentangle the two effects. As an application, we show that the empirical overlap observed in the International Trade Multiplex genuinely requires a nonzero inter-layer coupling in its modeling, as it is not merely a spurious result of the correlation between node degrees across different layers.

Vital node identification based on cycle structure in a multiplex network

ABSTRACT

Multiplex networks frame the heterogeneous nature of real systems, where the multiple roles of nodes, both functionally and structurally, are well represented. We identify these vital nodes in a multiplex network so that we can control a pandemic outbreak like COVID-19, eliminate damage from a network attack, maintain traffic, and so on. Vital node identification has attracted scientists in various fields for decades. In this paper, we propose a hybrid supra-cycle number and hybrid supra-cycle ratio based on the cycle structure, and present an extensive experimental analysis by comparing our indexes and several different indexes in four real multiplex networks on layer nodes and multiplex nodes. The experimental results show that these proposed indexes have good robustness, synchronization, and transmission dynamics. Finally, we provide an in-depth understanding of multiplex networks and cycle structure, and we sincerely hope more valuable academic achievements are proposed in the future.

Multiplex reconstruction with partial information

A multiplex is a collection of network layers, each representing a specific type of edges. This appears to be a genuine representation of many real-world systems. However, due to a variety of potential factors, such as limited budget and equipment, or physical impossibility, multiplex data can be difficult to observe directly. Often, only partial information on the layer structure of the system is available, whereas the remaining information is in the form of a single-layer network. In this work we face the problem of reconstructing the hidden multiplex structure of an aggregated network from partial information. We propose an algorithm that leverages the layerwise community structure that can be learned from partial observations to reconstruct the ground-truth topology of the unobserved part of the multiplex. The algorithm is characterized by a computational time that grows linearly with the network size. We perform a systematic study of reconstruction problems for both synthetic and real-world multiplex networks. We show that the ability of the proposed method to solve the reconstruction problem is affected by the heterogeneity of the individual layers and the similarity among the layers. On real-world networks, we observe that the accuracy of the reconstruction saturates quickly as the amount of available information increases. In genetic interaction and scientific collaboration multiplexes, for example, we find that 10% of ground-truth information yields 70% accuracy, while 30% information allows for more than 90% accuracy.

Influence of Information Blocking on the Spread of Virus in Multilayer Networks

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 affects the number of infected individuals.

Nodal-statistics-based equivalence relation for graph collections

Node role explainability in complex networks is very difficult yet is crucial in different application domains such as social science, neurosciences, or computer science. Many efforts have been made on the quantification of hubs revealing particular nodes in a network using a given structural property. Yet, in several applications, when multiple instances of networks are available and several structural properties appear to be relevant, the identification of node roles remains largely unexplored. Inspired by the node automorphically equivalence relation, we define an equivalence relation on graph nodes associated with any collection of nodal statistics (i.e., any functions on the node set). This allows us to define new graph global measures: the power coefficient and the orthogonality score to evaluate the parsimony and heterogeneity of a given nodal statistics collection. In addition, we introduce a new method based on structural patterns to compare graphs that have the same vertices set. This method assigns a value to a node to determine its role distinctiveness in a graph family. Extensive numerical results of our method are conducted on both generative graph models and real data concerning human brain functional connectivity. The differences in nodal statistics are shown to be dependent on the underlying graph structure. Comparisons between generative models and real networks combining two different nodal statistics reveal the complexity of human brain functional connectivity with differences at both global and nodal levels. Using a group of 200 healthy controls connectivity networks, our method computes high correspondence scores among the whole population to detect homotopy and finally quantify differences between comatose patients and healthy controls.

Multimodal multilayer network centrality relates to executive functioning

Executive functioning (EF) is a higher order cognitive process that is thought to depend on a network organization facilitating integration across subnetworks, in the context of which the central role of the fronto-parietal network (FPN) has been described across imaging and neurophysiological modalities. However, the potentially complementary unimodal information on the relevance of the FPN for EF has not yet been integrated. We employ a multilayer framework to allow for integration of different modalities into one 'network of networks.' We used diffusion MRI, resting-state functional MRI, MEG, and neuropsychological data obtained from 33 healthy adults to construct modality-specific single-layer networks as well as a single multilayer network per participant. We computed single-layer and multilayer eigenvector centrality of the FPN as a measure of integration in this network and examined their associations with EF. We found that higher multilayer FPN centrality, but not single-layer FPN centrality, was related to better EF. We did not find a statistically significant change in explained variance in EF when using the multilayer approach as compared to the single-layer measures. Overall, our results show the importance of FPN integration for EF and underline the promise of the multilayer framework toward better understanding cognitive functioning.

Modular structures and the delivery of inpatient care in hospitals: a Network Science perspective on healthcare function and dysfunction

BACKGROUND

Reinforced by the COVID-19 pandemic, the capacity of health systems to cope with increasing healthcare demands has been an abiding concern of both governments and the public. Health systems are made up from non-identical human and physical components interacting in diverse ways in varying locations. It is challenging to represent the function and dysfunction of such systems in a scientific manner. We describe a Network Science approach to that dilemma. General hospitals with large emergency caseloads are the resource intensive components of health systems. We propose that the care-delivery services in such entities are modular, and that their structure and function can be usefully analysed by contemporary Network Science. We explore that possibility in a study of Australian hospitals during 2019 and 2020.

METHODS

We accessed monthly snapshots of whole of hospital administrative patient level data in two general hospitals during 2019 and 2020. We represented the organisations inpatient services as network graphs and explored their graph structural characteristics using the Louvain algorithm and other methods. We related graph topological features to aspects of observable function and dysfunction in the delivery of care.

RESULTS

We constructed a series of whole of institution bipartite hospital graphs with clinical unit and labelled wards as nodes, and patients treated by units in particular wards as edges. Examples of the graphs are provided. Algorithmic identification of community structures confirmed the modular structure of the graphs. Their functional implications were readily identified by domain experts. Topological graph features could be related to functional and dysfunctional issues such as COVID-19 related service changes and levels of hospital congestion.

DISCUSSION AND CONCLUSIONS

Contemporary Network Science is one of the fastest growing areas of current scientific and technical advance. Network Science confirms the modular nature of healthcare service structures. It holds considerable promise for understanding function and dysfunction in healthcare systems, and for reconceptualising issues such as hospital capacity in new and interesting ways.

Examining COVID-19-triggered changes in spatial connectivity patterns in the European air transport network up to June 2021

The integrity of international supply chain operations heavily relies on air transport services to facilitate the movement of goods and enable human interactions between its stakeholders. With the outbreak of COVID-19 in Europe around March 2020, air transport networks have been subject to profound alterations. Although the link between variations in air transport service levels and changes in user costs for network-wide travel has been analysed extensively, few studies have examined the extent to which severe network shrinkage events lead to a reduction in network connectivity, which is therefore difficult to predict. This paper investigates how the COVID-19 pandemic has structurally altered the European air transport network in 2020/21 and how these changes have deteriorated users' ease when utilising network-wide air transport services. To do this, the paper estimates the change in average quickest path length at the airport level during different stages of this period. Results indicate there is strong heterogeneity in airports' susceptibility to pandemic-induced network changes, with both regional variations and variations in the airline type serving individual airports. Furthermore, topological features of individual airports are found to determine airport susceptibility. The findings are discussed in terms of their implications for locational decisions in supply chain designs.

Stability of synchronization in simplicial complexes with multiple interaction layers

Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in the presence of general coupling functions. It generalizes the well-known master stability function scheme to the higher-order structures with multiple interaction layers. We verify our theoretical results by employing them on networks of paradigmatic Rössler oscillators and Sherman neuronal models, and we demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.

Discrimination reveals reconstructability of multiplex networks from partial observations

An excellent method for predicting links in multiplex networks is reflected in its ability to reconstruct them accurately. Although link prediction methods perform well on estimating the existence probability of each potential link in monoplex networks by the set of partially observed links, we lack a mathematical tool to reconstruct the multiplex network from the observed aggregate topology and partially observed links in multiplex networks. Here, we fill this gap by developing a theoretical and computational framework that builds a probability space containing possible structures with a maximum likelihood estimation. Then, we discovered that the discrimination, an indicator quantifying differences between layers from an entropy perspective, determines the reconstructability, i.e., the accuracy of such reconstruction. This finding enables us to design the optimal strategy to allocate the set of observed links in different layers for promoting the optimal reconstruction of multiplex networks. Finally, the theoretical analyses are corroborated by empirical results from biological, social, engineered systems, and a large volume of synthetic networks.

Transportation Interrelation Embedded in Regional Development: The Characteristics and Drivers of Road Transportation Interrelation in Guangdong Province, China

Unbalanced regional development is often accompanied by a heterogeneity in regional transportation. The relationship between the interrelation of regional transportation and economic connections among cities remains unclear. This study attempts to explicate the structural characteristics of the spatial interrelation network of road transportation in Guangdong province. This study analyzes road traffic data in Guangdong province from 2015 to 2020 using a gravity model, social network analysis, and the quadratic assignment procedure (QAP). The results indicate that the spatial network of road transportation interrelations in Guangdong province have obvious core–periphery features. The intercity transportation interrelation in Guangdong province is significantly correlated with the differences in population density, vehicle population, and tourism resources, as well as the distance between cities; however, the effects of these factors vary across different regions. To promote balanced regional development, Guangdong province should strengthen the transportation interrelation between peripheral cities and other cities to raise the position of peripheral cities in the network. Introducing the required personnel and developing tourism resources with regional features would help develop peripheral cities that have a low population density and abundant tourism resources. This provincial transportation development strategy should consider balancing the development of mega metropolitan areas and non-coastal, small- and medium-sized cities to balance regional development.

Systematic assessment of the quality of fit of the stochastic block model for empirical networks

We perform a systematic analysis of the quality of fit of the stochastic block model (SBM) for 275 empirical networks spanning a wide range of domains and orders of size magnitude. We employ posterior predictive model checking as a criterion to assess the quality of fit, which involves comparing networks generated by the inferred model with the empirical network, according to a set of network descriptors. We observe that the SBM is capable of providing an accurate description for the majority of networks considered, but falls short of saturating all modeling requirements. In particular, networks possessing a large diameter and slow-mixing random walks tend to be badly described by the SBM. However, contrary to what is often assumed, networks with a high abundance of triangles can be well described by the SBM in many cases. We demonstrate that simple network descriptors can be used to evaluate whether or not the SBM can provide a sufficiently accurate representation, potentially pointing to possible model extensions that can systematically improve the expressiveness of this class of models.

Non-Markovian random walks characterize network robustness to nonlocal cascades

Localized perturbations in a real-world network have the potential to trigger cascade failures at the whole system level, hindering its operations and functions. Standard approaches analytically tackling this problem are mostly based either on static descriptions, such as percolation, or on models where the failure evolves through first-neighbor connections, crucially failing to capture the nonlocal behavior typical of real cascades. We introduce a dynamical model that maps the failure propagation across the network to a self-avoiding random walk that, at each step, has a probability to perform nonlocal jumps toward operational systems' units. Despite the inherent non-Markovian nature of the process, we are able to characterize the critical behavior of the system out of equilibrium, as well as the stopping time distribution of the cascades. Our numerical experiments on synthetic and empirical biological and transportation networks are in excellent agreement with theoretical expectation, demonstrating the ability of our framework to quantify the vulnerability to nonlocal cascade failures of complex systems with interconnected structure.

Impact of Centrality Measures on the Common Neighbors in Link Prediction for Multiplex Networks

Complex networks are representations of real-world systems that can be better modeled as multiplex networks, where the same nodes develop multi-type connections. One of the important concerns about these networks is link prediction, which has many applications in social networks and recommender systems. In this article, similarity-based methods such as common neighbors (CNs) are the mainstream. However, in the CN method, the contribution of each CN in the likelihood of new connections is equally taken into account. In this work, we propose a new link prediction method namely Weighted Common Neighbors (WCN), which is based on CNs and various types of Centrality measures (including degree, k-core, closeness, betweenness, Eigenvector, and PageRank) to predict the formation of new links in multiplex networks. So, in this model, each CN has a different impact on the node connection likelihood. Moreover, we investigate the impact of interlayer information on improving the performance of link prediction in the target layer. Using Area under the ROC Curve and precision as evaluation metrics, we perform a comprehensive experimental evaluation of our proposed method on seven real multiplex networks. The results validate the improved performance of our proposed method compared with existing methods, and we show that the performance of proposed methods is significantly improved while using interlayer information in multiplex networks.

Intralayer and interlayer synchronization in multiplex network with higher-order interactions

Recent developments in complex systems have witnessed that many real-world scenarios, successfully represented as networks, are not always restricted to binary interactions but often include higher-order interactions among the nodes. These beyond pairwise interactions are preferably modeled by hypergraphs, where hyperedges represent higher-order interactions between a set of nodes. In this work, we consider a multiplex network where the intralayer connections are represented by hypergraphs, called the multiplex hypergraph. The hypergraph is constructed by mapping the maximal cliques of a scale-free network to hyperedges of suitable sizes. We investigate the intralayer and interlayer synchronizations of such multiplex structures. Our study unveils that the intralayer synchronization appreciably enhances when a higher-order structure is taken into consideration in spite of only pairwise connections. We derive the necessary condition for stable synchronization states by the master stability function approach, which perfectly agrees with the numerical results. We also explore the robustness of interlayer synchronization and find that for the multiplex structures with many-body interaction, the interlayer synchronization is more persistent than the multiplex networks with solely pairwise interaction.

Fast computation of matrix function-based centrality measures for layer-coupled multiplex networks

Centrality measures identify and rank the most influential entities of complex networks. In this paper, we generalize matrix function-based centrality measures, which have been studied extensively for single-layer and temporal networks in recent years to layer-coupled multiplex networks. The layers of these networks can reflect different relationships and interactions between entities or changing interactions over time. We use the supra-adjacency matrix as network representation, which has already been used to generalize eigenvector centrality to temporal and multiplex networks. With a suitable choice of edge weights, the definition of single-layer matrix function-based centrality measures in terms of walks on networks carries over naturally to the multilayer case. In contrast to other walk-based centralities, matrix function-based centralities are parameterized measures, which have been shown to interpolate between (local) degree and (global) eigenvector centrality in the single-layer case. As the explicit evaluation of the involved matrix function expressions becomes infeasible for medium to large-scale networks, we present highly efficient approximation techniques from numerical linear algebra, which rely on Krylov subspace methods, Gauss quadrature, and stochastic trace estimation. We present extensive numerical studies on synthetic and real-world multiplex transportation, communication, and collaboration networks. The comparison with established multilayer centrality measures shows that our framework produces meaningful rankings of nodes, layers, and node-layer pairs. Furthermore, our experiments corroborate the linear computational complexity of the employed numerical methods in terms of the network size that is theoretically indicated under the assumption of sparsity in the supra-adjacency matrix. This excellent scalability allows the efficient treatment of large-scale networks with the number of node-layer pairs of order 10^{7} or higher.

Analysis of the Structure and Dynamics of European Flight Networks

We analyze structure and dynamics of flight networks of 50 airlines active in the European airspace in 2017. Our analysis shows that the concentration of the degree of nodes of different flight networks of airlines is markedly heterogeneous among airlines reflecting heterogeneity of the airline business models. We obtain an unsupervised classification of airlines by performing a hierarchical clustering that uses a correlation coefficient computed between the average occurrence profiles of 4-motifs of airline networks as similarity measure. The hierarchical tree is highly informative with respect to properties of the different airlines (for example, the number of main hubs, airline participation to intercontinental flights, regional coverage, nature of commercial, cargo, leisure or rental airline). The 4-motif patterns are therefore distinctive of each airline and reflect information about the main determinants of different airlines. This information is different from what can be found looking at the overlap of directed links.

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.

Stability analysis of intralayer synchronization in time-varying multilayer networks with generic coupling functions

The stability analysis of synchronization patterns on generalized network structures is of immense importance nowadays. In this article, we scrutinize the stability of intralayer synchronous state in temporal multilayer hypernetworks, where each dynamic units in a layer communicate with others through various independent time-varying connection mechanisms. Here, dynamical units within and between layers may be interconnected through arbitrary generic coupling functions. We show that intralayer synchronous state exists as an invariant solution. Using fast-switching stability criteria, we derive the condition for stable coherent state in terms of associated time-averaged network structure, and in some instances we are able to separate the transverse subspace optimally. Using simultaneous block diagonalization of coupling matrices, we derive the synchronization stability condition without considering time-averaged network structure. Finally, we verify our analytically derived results through a series of numerical simulations on synthetic and real-world neuronal networked systems.

Link overlap influences opinion dynamics on multiplex networks of Ashkin-Teller spins

Consider a multiplex network formed by two layers indicating social interactions: the first layer is a friendship network and the second layer is a network of business relations. In this duplex network each pair of individuals can be connected in different ways: they can be connected by a friendship but not connected by a business relation, they can be connected by a business relation without being friends, or they can be simultaneously friends and in a business relation. In the latter case we say that the links in different layers overlap. These three types of connections are called multilinks and the multidegree indicates the sum of multilinks of a given type that are incident to a given node. Previous opinion models on multilayer networks have mostly neglected the effect of link overlap. Here we show that link overlap can have important effects in the formation of a majority opinion. Indeed, the formation of a majority opinion can be significantly influenced by the statistical properties of multilinks, and in particular by the multidegree distribution. To quantitatively address this problem, we study a simple spin model, called the Ashkin-Teller model, including two-body and four-body interactions between nodes in different layers. Here we fully investigate the rich phase diagram of this model which includes a large variety of phase transitions. Indeed, the phase diagram or the model displays continuous, discontinuous, and hybrid phase transitions, and successive jumps of the order parameters within the Baxter phase.

The Evolvement of Rail Transit Network Structure and Impact on Travel Characteristics: A Case Study of Wuhan

The expansion of the rail transit network has a positive impact on travel characteristics under spatial and temporal constraints by changing accessibility. However, few empirical studies have examined the longitudinal evolution of the impact of accessibility and travel characteristics. In this paper, a model of the Wuhan rail transit network is constructed and the evolution of the spatial pattern of accessibility over different periods is analyzed. The correlation of accessibility with rail transit travel characteristics is studied longitudinally to provide theoretical support for rail transit construction and traffic demand management. The study shows that: (1) Wuhan’s rail transit network has evolved from a tree to a ring, improving the operational efficiency. (2) The accessibility of Wuhan’s rail transit network has evolved into a circular structure, showing a decreasing trend away from the city center. (3) The change of accessibility greatly affects travel characteristics. The higher the accessibility, the higher the traffic volume, and the lower the travel frequency, the more residents travel during peak hours, and the shorter the travel distance. These findings are useful for gaining insight into public transportation demand in large cities, and thus for developing reasonable transportation demand management policies.

Measuring the effect of distance on the network topology of the Global Container Shipping Network

This paper examines how spatial distance affects network topology on empirical data concerning the Global Container Shipping Network (GCSN). The GCSN decomposes into 32 multiplex layers, defined at several spatial levels, by successively removing connections of smaller distances. This multilayer decomposition approach allows studying the topological properties of each layer as a function of distance. The analysis provides insights into the hierarchical structure and (importing and exporting) trade functionality of the GCSN, hub connectivity, several topological aspects, and the distinct role of China in the network's structure. It also shows that bidirectional links decrease with distance, highlighting the importance of asymmetric functionality in carriers' operations. It further configures six novel clusters of ports concerning their spatial coverage. Finally, it reveals three levels of geographical scale in the structure of GCSN (where the network topology significantly changes): the neighborhood (local connectivity); the scale of international connectivity (mesoscale or middle connectivity); and the intercontinental market (large scale connectivity). The overall approach provides a methodological framework for analyzing network topology as a function of distance, highlights the spatial dimension in complex and multilayer networks, and provides insights into the spatial structure of the GCSN, which is the most important market of the global maritime economy.

Structural and spectral properties of generative models for synthetic multilayer air transportation networks

To understand airline transportation networks (ATN) systems we can effectively represent them as multilayer networks, where layers capture different airline companies, the nodes correspond to the airports and the edges to the routes between the airports. We focus our study on the importance of leveraging synthetic generative multilayer models to support the analysis of meaningful patterns in these routes, capturing an ATN’s evolution with an emphasis on measuring its resilience to random or targeted attacks and considering deliberate locations of airports. By resorting to the European ATN and the United States ATN as exemplary references, in this work, we provide a systematic analysis of major existing synthetic generation models for ATNs, specifically ANGEL, STARGEN and BINBALL. Besides a thorough study of the topological aspects of the ATNs created by the three models, our major contribution lays on an unprecedented investigation of their spectral characteristics based on Random Matrix Theory and on their resilience analysis based on both site and bond percolation approaches. Results have shown that ANGEL outperforms STARGEN and BINBALL to better capture the complexity of real-world ATNs by featuring the unique properties of building a multiplex ATN layer by layer and of replicating layers with point-to-point structures alongside hub-spoke formations.

Multilayer network structure enhances the coexistence of competitive species

The concept of a multiplex network can be used to characterize the dispersal paths and states of different species in a patch habitat system. The multiplex network is one of three types of multilayer networks. In this study, the effect of a multiplex network on the long-term stable coexistence of species is investigated using the concept of metapopulation. Based on the mean field approximation, the stability analysis of a two-species system shows that, compared to the single layer network, the multiplex network is more conducive to the stable coexistence of species when one species has a stronger colonization ability. That is, in such a patch habitat system, if the dispersal paths of the stronger species are different than those of the weaker species, then the larger the heterogeneity of the dispersal network of the stronger species is, the more likely the long-term stable coexistence of species. This result provides a different perspective for understanding the biodiversity in heterogeneous habitats.

Calcarine as a bridge between brain function and structure in irritable bowel syndrome: A multiplex network analysis

BACKGROUND AND AIM

Jointly analyzing structural and functional brain networks enables a better understanding of pathological underpinnings of irritable bowel syndrome (IBS). Multiplex network analysis provides a novel framework to study complex networks consisting of different types of connectivity patterns in multimodal data.

METHODS

In the present work, we integrated functional and structural networks to a multiplex network. Then, the multiplex metrics and the inner-layer/inter-layer hub nodes were investigated through 34 patients with IBS and 33 healthy controls.

RESULTS

Significantly differential multiplex degree in both left and right parts of calcarine was found, and meanwhile, IBS patients lost inner-layer hub properties in these regions. In addition, the left fusiform was no longer practicing as an inner-layer hub node, while the right median cingulate acted as a new inner-layer hub node in the IBS patients. Besides, the right calcarine, which lost its inner-layer hub identity, became a new inter-layer hub node, and the multiplex degree of the left hippocampus, which lost its inter-layer hub identity in IBS patients, was significantly positively correlated with the IBS Symptom Severity Score scores.

CONCLUSIONS

Inner-layer hub nodes of multiplex networks were preferentially vulnerable, and some inner-layer hub nodes would convert into inter-layer hub nodes in IBS patients. Besides, the inter-layer hub nodes might be influenced by IBS severity and therefore converted to general nodes.

An adaptive node embedding framework for multiplex networks

Network Embedding (NE) has emerged as a powerful tool in many applications. Many real-world networks have multiple types of relations between the same entities, which are appropriate to be modeled as multiplex networks. However, at random walk-based embedding study for multiplex networks, very little attention has been paid to the problems of sampling bias and imbalanced relation types. In this paper, we propose an Adaptive Node Embedding Framework (ANEF) based on cross-layer sampling strategies of nodes for multiplex networks. ANEF is the first framework to focus on the bias issue of sampling strategies. Through metropolis hastings random walk (MHRW) and forest fire sampling (FFS), ANEF is less likely to be trapped in local structure with high degree nodes. We utilize a fixed-length queue to record previously visited layers, which can balance the edge distribution over different layers in sampled node sequence processes. In addition, to adaptively sample the cross-layer context of nodes, we also propose a node metric called Neighbors Partition Coefficient (NPC). Experiments on real-world networks in diverse fields show that our framework outperforms the state-of-the-art methods in application tasks such as cross-domain link prediction and mutual community detection.

Antiphase synchronization in multiplex networks with attractive and repulsive interactions

A series of recent publications, within the framework of network science, have focused on the coexistence of mixed attractive and repulsive (excitatory and inhibitory) interactions among the units within the same system, motivated by the analogies with spin glasses as well as to neural networks, or ecological systems. However, most of these investigations have been restricted to single layer networks, requiring further analysis of the complex dynamics and particular equilibrium states that emerge in multilayer configurations. This article investigates the synchronization properties of dynamical systems connected through multiplex architectures in the presence of attractive intralayer and repulsive interlayer connections. This setting enables the emergence of antisynchronization, i.e., intralayer synchronization coexisting with antiphase dynamics between coupled systems of different layers. We demonstrate the existence of a transition from interlayer antisynchronization to antiphase synchrony in any connected bipartite multiplex architecture when the repulsive coupling is introduced through any spanning tree of a single layer. We identify, analytically, the required graph topologies for interlayer antisynchronization and its interplay with intralayer and antiphase synchronization. Next, we analytically derive the invariance of intralayer synchronization manifold and calculate the attractor size of each oscillator exhibiting interlayer antisynchronization together with intralayer synchronization. The necessary conditions for the existence of interlayer antisynchronization along with intralayer synchronization are given and numerically validated by considering Stuart-Landau oscillators. Finally, we also analytically derive the local stability condition of the interlayer antisynchronization state using the master stability function approach.

Sequential seeding in multilayer networks

Multilayer networks are the underlying structures of multiple real-world systems where we have more than one type of interaction/relation between nodes: social, biological, computer, or communication, to name only a few. In many cases, they are helpful in modeling processes that happen on top of them, which leads to gaining more knowledge about these phenomena. One example of such a process is the spread of influence. Here, the members of a social system spread the influence across the network by contacting each other, sharing opinions or ideas, or-explicitly-by persuasion. Due to the importance of this process, researchers investigate which members of a social network should be chosen as initiators of influence spread to maximize the effect. In this work, we follow this direction and develop and evaluate the sequential seeding technique for multilayer networks. Until now, such techniques were evaluated only using simple one layer networks. The results show that sequential seeding in multilayer networks outperforms the traditional approach by increasing the coverage and allowing to save the seeding budget. However, it also extends the duration of the spreading process.

Inference of edge correlations in multilayer networks

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 approach 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 the connectivity patterns in different layers. Finally, we explain how to use correlated models for edge "prediction" (i.e., inference) in multilayer networks. By incorporating edge correlations, we find that prediction accuracy improves both in synthetic networks and in a temporal network of shoppers who are connected to previously purchased grocery products.

Dynamic Analysis of Muscle Coordination at Different Force Levels during Grip and Pinch with Multiplex Recurrence Network

Muscle synergistic contraction to produce force has been recognized as an important neurophysiological mechanism in neuromuscular system. Despite a range of approaches, such as nonnegative matrix factorization or principal component analysis that have been widely used, limitations still exist in analysis of dynamic coordination of multiple muscles. In addition, it is still less studied about the potential difference of muscle dynamic coordination at different force levels during grip and pinch within the same framework. With this aim, this study analyzed the dynamic coordination of multiple upper-limb muscles at low, medium and high force levels during pinch and grip with multiplex recurrence network (MRN). Twenty-four healthy subjects participated in the experiment. Subjects were instructed to grip an apparatus to match the target force as stably as they could for 10 s. Surface electromyographic (sEMG) signals were recorded from 8 upper-limb muscles and analyzed by the MRN. The interlayer mutual information (I) and the average edge overlap (ω) of MRNs were calculated to quantify muscle correlation and muscle synchronization, respectively. Results showed that I and ω values of extrinsic muscles' MRNs during grip were significantly higher than that during grip at medium and high force. Furthermore, the I and ω values of extrinsic muscle networks during grip increased with augmented force levels. No significant changes were found for the intrinsic muscles with force output levels. These findings indicate that the muscles coordination patterns between grip and pinch were different and higher co-contraction of extrinsic muscles is favorable to synergistic force production. With the force output increased, muscles' coordination was augmented in extrinsic muscles, but no change in intrinsic muscles because of independent and complicated control of fingers. This study provides an analytical tool for dynamic muscles coordination and provides insights into the mechanisms of synergistic control of muscle contractions for force production.Clinical Relevance-This study provides a novel analytical tool for muscle coordination during force production, which may facilitate the evaluation of neuromuscular function or serve as indicators for neuromuscular disorders.

Building a trust-based doctor recommendation system on top of multilayer graph database

In healthcare applications, developing a data model for storing patient-doctor relationships is important. Though relational models are popular for many commercial and business applications, they may not be appropriate for modeling patient-doctor relationships due to their inherent irregular nature and complexities. In this paper, as a case study, we propose to build a doctor recommendation system for the patients. The recommendation system is built on top of a multilayer graph data model. Contemporary research papers have already shown that multilayer graph data models can be efficiently used in many applications where large, heterogeneous data are to be modeled. As part of the recommendation system, the paper also introduces a concept of trust which is one important ingredient of any kind of recommendation. The trust factor introduced in the paper exploits certain characteristics of the multilayer graph model. The paper also presents some analysis to demonstrate the efficiency of the graph data model in comparison with relational data model.

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.

Symmetries and cluster synchronization in multilayer networks

Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries, which occur in one layer and are independent of the other layers, and dependent layer symmetries, which involve nodes in different layers. We study stability of the cluster synchronous solution by decoupling the problem into a number of independent blocks and assessing stability of each block through a Master Stability Function. We see that blocks associated with dependent layer symmetries have a different structure to the other blocks, which affects the stability of clusters associated with these symmetries. Finally, we validate the theory in a fully analog experiment in which seven electronic oscillators of three kinds are connected with two kinds of coupling.

Identifying Influencers in Social Networks

Social network analysis is a multidisciplinary research covering informatics, mathematics, sociology, management, psychology, etc. In the last decade, the development of online social media has provided individuals with a fascinating platform of sharing knowledge and interests. The emergence of various social networks has greatly enriched our daily life, and simultaneously, it brings a challenging task to identify influencers among multiple social networks. The key problem lies in the various interactions among individuals and huge data scale. Aiming at solving the problem, this paper employs a general multilayer network model to represent the multiple social networks, and then proposes the node influence indicator merely based on the local neighboring information. Extensive experiments on 21 real-world datasets are conducted to verify the performance of the proposed method, which shows superiority to the competitors. It is of remarkable significance in revealing the evolutions in social networks and we hope this work will shed light for more and more forthcoming researchers to further explore the uncharted part of this promising field.

Visibility Graphs for Image Processing

The family of image visibility graphs (IVG/IHVGs) have been recently introduced as simple algorithms by which scalar fields can be mapped into graphs. Here we explore the usefulness of such\an operator in the scenario of image processing and image classification. We demonstrate that the link architecture of the image visibility graphs encapsulates relevant information on the structure of the images and we explore their potential as image filters. We introduce several graph features, including the novel concept of Visibility Patches, and show through several examples that these features are highly informative, computationally efficient and universally applicable for general pattern recognition and image classification tasks.

Invariance and stability conditions of interlayer synchronization manifold

We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two types of connections, one is the intralayer connection in each layer with hypernetwork structure and the other is the interlayer connection between the layers. Here all types of interactions within and between the layers are allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for local stability of the interlayer synchronization state using master stability function approach and a sufficient condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections. Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.

Grade Setting of a Timber Logistics Center Based on a Complex Network: A Case Study of 47 Timber Trading Markets in China

The location and grade setting of a timber logistics center is an important link in the optimization of the timber logistics system, the rationality of which can effectively improve the efficiency of the timber logistics supply chain. There is a long distance between the main forested areas in China, and more than 55% of the timber demand depends on imports. Research and practice of systematically planning timber logistics centers in the whole country have not been well carried out, which reduces the efficiency of timber logistics. In this paper, 47 timber trading markets with a certain scale in China are selected as the basis for logistics center selection. Based on their transportation network relationship and the number of enterprises in the market, combined with the complex network theory and data analysis method, the network characteristics of three different transportation networks are measured. After determining the transportation capacity indicator, the logistics capacity coefficient is measured based on the freight volume of each node. Then, the important nodes are identified, and each node is graded to systematically set up the timber logistics center.

Time evolution of the behaviour of Brazilian legislative Representatives using a complex network approach

The follow up of Representative behavior after elections is imperative for a democratic Representative system, at the very least to punish betrayal with no re-election. Our goal was to show how to follow Representatives’ and how to show behavior in real situations and observe trends in political crises including the onset of game changing political instabilities. We used correlation and correlation distance matrices of Brazilian Representative votes during four presidential terms. Re-ordering these matrices with Minimal Spanning Trees displays the dynamical formation of clusters for the sixteen year period, which includes one Presidential impeachment. The reordered matrices, colored by correlation strength and by the parties clearly show the origin of observed clusters and their evolution over time. When large clusters provide government support cluster breaks, political instability arises, which could lead to an impeachment, a trend we observed three years before the Brazilian President was impeached. We believe this method could be applied to foresee other political storms.