Growing multiplex networks

Dynamics of cascades in spatial interdependent networks

The dynamics of cascading failures in spatial interdependent networks significantly depends on the interaction range of dependency couplings between layers. In particular, for an increasing range of dependency couplings, different types of phase transition accompanied by various cascade kinetics can be observed, including mixed-order transition characterized by critical branching phenomena, first-order transition with nucleation cascades, and continuous second-order transition with weak cascades. We also describe the dynamics of cascades at the mutual mixed-order resistive transition in interdependent superconductors and show its similarity to that of percolation of interdependent abstract networks. Finally, we lay out our perspectives for the experimental observation of these phenomena, their phase diagrams, and the underlying kinetics, in the context of physical interdependent networks. Our studies of interdependent networks shed light on the possible mechanisms of three known types of phase transitions, second order, first order, and mixed order as well as predicting a novel fourth type where a microscopic intervention will yield a macroscopic phase transition.

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.

Fractal Fluctuations at Mixed-Order Transitions in Interdependent Networks

We study the critical features of the order parameter's fluctuations near the threshold of mixed-order phase transitions in randomly interdependent spatial networks. Remarkably, we find that although the structure of the order parameter is not scale invariant, its fluctuations are fractal up to a well-defined correlation length ξ^{'} that diverges when approaching the mixed-order transition threshold. We characterize the self-similar nature of these critical fluctuations through their effective fractal dimension d_{f}^{'}=3d/4, and correlation length exponent ν^{'}=2/d, where d is the dimension of the system. By analyzing percolation and magnetization, we demonstrate that d_{f}^{'} and ν^{'} are the same for both, i.e., independent of the symmetry of the process for any d of the underlying networks.

A tensor-based formulation of hetero-functional graph theory

Recently, hetero-functional graph theory (HFGT) has developed as a means to mathematically model the structure of large-scale complex flexible engineering systems. It does so by fusing concepts from network science and model-based systems engineering (MBSE). For the former, it utilizes multiple graph-based data structures to support a matrix-based quantitative analysis. For the latter, HFGT inherits the heterogeneity of conceptual and ontological constructs found in model-based systems engineering including system form, system function, and system concept. These diverse conceptual constructs indicate multi-dimensional rather than two-dimensional relationships. This paper provides the first tensor-based treatment of hetero-functional graph theory. In particular, it addresses the "system concept" and the hetero-functional adjacency matrix from the perspective of tensors and introduces the hetero-functional incidence tensor as a new data structure. The tensor-based formulation described in this work makes a stronger tie between HFGT and its ontological foundations in MBSE. Finally, the tensor-based formulation facilitates several analytical results that provide an understanding of the relationships between HFGT and multi-layer networks.

Visibility graph for time series prediction and image classification: a review

The analysis of time series and images is significant across different fields due to their widespread applications. In the past few decades, many approaches have been developed, including data-driven artificial intelligence methods, mechanism-driven physical methods, and hybrid mechanism and data-driven models. Complex networks have been used to model numerous complex systems due to its characteristics, including time series prediction and image classification. In order to map time series and images into complex networks, many visibility graph algorithms have been developed, such as horizontal visibility graph, limited penetrable visibility graph, multiplex visibility graph, and image visibility graph. The family of visibility graph algorithms will construct different types of complex networks, including (un-) weighted, (un-) directed, and (single-) multi-layered networks, thereby focusing on different kinds of properties. Different types of visibility graph algorithms will be reviewed in this paper. Through exploring the topological structure and information in the network based on statistical physics, the property of time series and images can be discovered. In order to forecast (multivariate) time series, several variations of local random walk algorithms and different information fusion approaches are applied to measure the similarity between nodes in the network. Different forecasting frameworks are also proposed to consider the information in the time series based on the similarity. In order to classify the image, several machine learning models (such as support vector machine and linear discriminant) are used to classify images based on global features, local features, and multiplex features. Through various simulations on a variety of datasets, researchers have found that the visibility graph algorithm outperformed existing algorithms, both in time series prediction and image classification. Clearly, complex networks are closely connected with time series and images by visibility graph algorithms, rendering complex networks to be an important tool for understanding the characteristics of time series and images. Finally, we conclude in the last section with future outlooks for the visibility graph.

Pattern Discovery in Multilayer Networks

MOTIVATION

In bioinformatics, complex cellular modeling and behavior simulation to identify significant molecular interactions is considered a relevant problem. Traditional methods model such complex systems using single and binary network. However, this model is inadequate to represent biological networks as different sets of interactions can simultaneously take place for different interaction constraints (such as transcription regulation and protein interaction). Furthermore, biological systems may exhibit varying interaction topologies even for the same interaction type under different developmental stages or stress conditions. Therefore, models which consider biological systems as solitary interactions are inaccurate as they fail to capture the complex behavior of cellular interactions within organisms. Identification and counting of recurrent motifs within a network is one of the fundamental problems in biological network analysis. Existing methods for motif counting on single network topologies are inadequate to capture patterns of molecular interactions that have significant changes in biological expression when identified across different organisms that are similar, or even time-varying networks within the same organism. That is, they fail to identify recurrent interactions as they consider a single snapshot of a network among a set of multiple networks. Therefore, we need methods geared towards studying multiple network topologies and the pattern conservation among them. Contributions: In this paper, we consider the problem of counting the number of instances of a user supplied motif topology in a given multilayer network. We model interactions among a set of entities (e.g., genes)describing various conditions or temporal variation as multilayer networks. Thus a separate network as each layer shows the connectivity of the nodes under a unique network state. Existing motif counting and identification methods are limited to single network topologies, and thus cannot be directly applied on multilayer networks. We apply our model and algorithm to study frequent patterns in cellular networks that are common in varying cellular states under different stress conditions, where the cellular network topology under each stress condition describes a unique network layer.

RESULTS

We develop a methodology and corresponding algorithm based on the proposed model for motif counting in multilayer networks. We performed experiments on both real and synthetic datasets. We modeled the synthetic datasets under a wide spectrum of parameters, such as network size, density, motif frequency. Results on synthetic datasets demonstrate that our algorithm finds motif embeddings with very high accuracy compared to existing state-of-the-art methods such as G-tries, ESU (FANMODE)and mfinder. Furthermore, we observe that our method runs from several times to several orders of magnitude faster than existing methods. For experiments on real dataset, we consider Escherichia coli (E. coli)transcription regulatory network under different experimental conditions. We observe that the genes selected by our method conserves functional characteristics under various stress conditions with very low false discovery rates. Moreover, the method is scalable to real networks in terms of both network size and number of layers.

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.

Persistence in multilayer ecological network consisting of harvested patches

Complex network theory yields a powerful approach to solve the difficulties arising in a major section of ecological systems, prey-predator interaction being one among them. A large variety of ecological systems have been successfully investigated employing the theory of complex networks, and one of the most significant advancements in this theory is the emerging field of multilayer networks. The field of multilayer networks provides a natural framework to accommodate multiple layers of complexities emerging in ecosystems. In this article, we consider prey-predator patches communicating among themselves while being connected by distinct small-world dispersal topologies in two layers of the network. We scrutinize the robustness of the multilayer ecological network sustaining gradually over harvested patches. We thoroughly report the consequences of introducing asymmetries in both interlayer and intralayer dispersal strengths as well as the network topologies on the global persistence of species in the network. Besides numerical simulation, we analytically derive the critical point up to which the network can sustain species in the network. Apart from the results on a purely multiplex framework, we validate our claims for multilayer formalism in which the patches of the layers are different. Interestingly, we observe that due to the interaction between the two layers, species are recovered in the layer that we assume to be extinct initially. Moreover, we find similar results while considering two completely different prey-predator systems, which eventually attests that the outcomes are not model specific.

The structure of co-publications multilayer network

Using the headers of scientific papers, we have built multilayer networks of entities involved in research namely: authors, laboratories, and institutions. We have analyzed some properties of such networks built from data extracted from the HAL archives and found that the network at each layer is a small-world network with power law distribution. In order to simulate such co-publication network, we propose a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers. The clique is built from new and existing nodes selected using preferential attachment. We also show that, the degree distribution of generated layers follows a power law. From the simulations of our model, we show that the generated multilayer networks reproduce the studied properties of co-publication networks.

Epidemic spreading and control strategies in spatial modular network

Epidemic spread on networks is one of the most studied dynamics in network science and has important implications in real epidemic scenarios. Nonetheless, the dynamics of real epidemics and how it is affected by the underline structure of the infection channels are still not fully understood. Here we apply the susceptible-infected-recovered model and study analytically and numerically the epidemic spread on a recently developed spatial modular model imitating the structure of cities in a country. The model assumes that inside a city the infection channels connect many different locations, while the infection channels between cities are less and usually directly connect only a few nearest neighbor cities in a two-dimensional plane. We find that the model experience two epidemic transitions. The first lower threshold represents a local epidemic spread within a city but not to the entire country and the second higher threshold represents a global epidemic in the entire country. Based on our analytical solution we proposed several control strategies and how to optimize them. We also show that while control strategies can successfully control the disease, early actions are essentials to prevent the disease global spread.

Multilayer network approach to modeling authorship influence on citation dynamics in physics journals

We provide a general framework to model the growth of networks consisting of different coupled layers. Our aim is to estimate the impact of one such layer on the dynamics of the others. As an application, we study a scientometric network, where one layer consists of publications as nodes and citations as links, whereas the second layer represents the authors. This allows us to address the question of how characteristics of authors, such as their number of publications or number of previous coauthors, impacts the citation dynamics of a new publication. To test different hypotheses about this impact, our model combines citation constituents and social constituents in different ways. We then evaluate their performance in reproducing the citation dynamics in nine different physics journals. For this, we develop a general method for statistical parameter estimation and model selection that is applicable to growing multilayer networks. It takes both the parameter errors and the model complexity into account and is computationally efficient and scalable to large networks.

Exploring Drug Treatment Patterns Based on the Action of Drug and Multilayer Network Model

Some drugs can be used to treat multiple diseases, suggesting potential patterns in drug treatment. Determination of drug treatment patterns can improve our understanding of the mechanisms of drug action, enabling drug repurposing. A drug can be associated with a multilayer tissue-specific protein-protein interaction (TSPPI) network for the diseases it is used to treat. Proteins usually interact with other proteins to achieve functions that cause diseases. Hence, studying drug treatment patterns is similar to studying common module structures in multilayer TSPPI networks. Therefore, we propose a network-based model to study the treatment patterns of drugs. The method was designated SDTP (studying drug treatment pattern) and was based on drug effects and a multilayer network model. To demonstrate the application of the SDTP method, we focused on analysis of trichostatin A (TSA) in leukemia, breast cancer, and prostate cancer. We constructed a TSPPI multilayer network and obtained candidate drug-target modules from the network. Gene ontology analysis provided insights into the significance of the drug-target modules and co-expression networks. Finally, two modules were obtained as potential treatment patterns for TSA. Through analysis of the significance, composition, and functions of the selected drug-target modules, we validated the feasibility and rationality of our proposed SDTP method for identifying drug treatment patterns. In summary, our novel approach used a multilayer network model to overcome the shortcomings of single-layer networks and combined the network with information on drug activity. Based on the discovered drug treatment patterns, we can predict the potential diseases that the drug can treat. That is, if a disease-related protein module has a similar structure, then the drug is likely to be a potential drug for the treatment of the disease.

Analysis of the Level of Service of Unloading Zones Using Diversity Measures in a Multiplex Network

Unloading zones are a fundamental part of the infrastructure of urban freight transport. The location and accessibility of unloading zones to commercial establishments reduce the operating time and, consequently, the transportation costs. In general, unloading zones are located on-street and allocated by local authorities. In this context, this paper aims to evaluate the level of service of unloading zones. The research approach uses the diversity measures in a multiplex network to identify the level of service and cargo accessibility of unloading zones. An analysis is developed for the central area of Belo Horizonte (Brazil). The results indicate that unloading zones located up to 25 m from the establishments have a high accessibility and low level of service. In contrast, unloading zones located up to 100 m from the establishments have a low accessibility and high level of service. These results allow us to conclude that the planning process of the location of unloading zones in Belo Horizonte is flawed. In addition, the maximum distance from unloading zones to establishments must be 75 m, so that there is a balance between the accessibility and level of service.

Generalized k-core percolation on correlated and uncorrelated multiplex networks

It has been recognized that multiplexes and interlayer degree correlations can play a crucial role in the resilience of many real-world complex systems. Here we introduce a multiplex pruning process that removes nodes of degree less than k_{i} and their nearest neighbors in layer i for i=1,...,m, and establish a generic framework of generalized k-core (Gk-core) percolation over interlayer uncorrelated and correlated multiplex networks of m layers, where k=(k_{1},...,k_{m}) and m is the total number of layers. Gk-core exhibits a discontinuous phase transition for all k owing to cascading failures. We have unraveled the existence of a tipping point of the number of layers, above which the Gk-core collapses abruptly. This dismantling effect of multiplexity on Gk-core percolation shows a diminishing marginal utility in homogeneous networks when the number of layers increases. Moreover, we have found the assortative mixing for interlayer degrees strengthens the Gk-core but still gives rise to discontinuous phase transitions as compared to the uncorrelated counterparts. Interlayer disassortativity on the other hand weakens the Gk-core structure. The impact of correlation effect on Gk-core tends to be more salient systematically over k for heterogenous networks than homogeneous ones.

A multiplex, multi-timescale model approach for economic and frequency control in power grids

Power systems are subject to fundamental changes due to the increasing infeed of decentralized renewable energy sources and storage. The decentralized nature of the new actors in the system requires new concepts for structuring the power grid and achieving a wide range of control tasks ranging from seconds to days. Here, we introduce a multiplex dynamical network model covering all control timescales. Crucially, we combine a decentralized, self-organized low-level control and a smart grid layer of devices that can aggregate information from remote sources. The safety-critical task of frequency control is performed by the former and the economic objective of demand matching dispatch by the latter. Having both aspects present in the same model allows us to study the interaction between the layers. Remarkably, we find that adding communication in the form of aggregation does not improve the performance in the cases considered. Instead, the self-organized state of the system already contains the information required to learn the demand structure in the entire grid. The model introduced here is highly flexible and can accommodate a wide range of scenarios relevant to future power grids. We expect that it is especially useful in the context of low-energy microgrids with distributed generation.

A Feasible Community Detection Algorithm for Multilayer Networks

As a more complicated network model, multilayer networks provide a better perspective for describing the multiple interactions among social networks in real life. Different from conventional community detection algorithms, the algorithms for multilayer networks can identify the underlying structures that contain various intralayer and interlayer relationships, which is of significance and remains a challenge. In this paper, aiming at the instability of the label propagation algorithm (LPA), an improved label propagation algorithm based on the SH-index (SH-LPA) is proposed. By analyzing the characteristics and deficiencies of the H-index, the SH-index is presented as an index to evaluate the importance of nodes, and the stability of the SH-LPA algorithm is verified by a series of experiments. Afterward, considering the deficiency of the existing multilayer network aggregation model, we propose an improved multilayer network aggregation model that merges two networks into a weighted single-layer network. Finally, considering the influence of the SH-index and the weight of the edge of the weighted network, a community detection algorithm (MSH-LPA) suitable for multilayer networks is exhibited in terms of the SH-LPA algorithm, and the superiority of the mentioned algorithm is verified by experimental analysis.

Temporal Network Pattern Identification by Community Modelling

Temporal network mining tasks are usually hard problems. This is because we need to face not only a large amount of data but also its non-stationary nature. In this paper, we propose a method for temporal network pattern representation and pattern change detection following the reductionist approach. The main idea is to model each stable (durable) state of a given temporal network as a community in a sampled static network and the temporal state change is represented by the transition from one community to another. For this purpose, a reduced static single-layer network, called a target network, is constructed by sampling and rearranging the original temporal network. Our approach provides a general way not only for temporal networks but also for data stream mining in topological space. Simulation results on artificial and real temporal networks show that the proposed method can group different temporal states into different communities with a very reduced amount of sampled nodes.

Organizational and Formational Structures of Networks in the Mental Lexicon: A State-Of-The-Art through Systematic Review

This state-of-the-art presents a systematic exploration on the use of network patterns in global research efforts to understand, organize and represent the mental lexicon. Results have shown an increase over recent years in the usage of complex, small-world and scale-free network patterns within the literature. With the increasing complexity of network patterns, we see more potential in the inter-disciplinary exploration of the mental lexicon through universal and mathematically-describable, behavioral patterns in small-world and scale-free networks. A systematic review of 36 items of methodologically-selected literature serve as a means to explore how the greater literary body understands network structures within the mental lexicon. Network-based approaches are discriminated between three contrasting varieties. These include: ‘simple networks’, characterized by arbitrarily organized graph patterns of metaphorical importance; ‘connectionist networks’, a broad category of networks which explore the structural features of a system through the analysis of emergent properties; and lastly ‘complex networks’, distinguished as small-world, scale-free networks which follow a strict and mathematically-describable structure in agreement with the Barabási–Albert model. Each network approach is explored in terms of their discernible differences which relate to their parameters and affect their implications. A final evaluation of observed patterns within the selected literature is offered, as well as an elaboration on the sense of trajectory beheld in the research in order to offer insight and orientation for future research.

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.

Deep Learning and Multiplex Networks for Accurate Modeling of Brain Age

Recent works have extensively investigated the possibility to predict brain aging from T1-weighted MRI brain scans. The main purposes of these studies are the investigation of subject-specific aging mechanisms and the development of accurate models for age prediction. Deviations between predicted and chronological age are known to occur in several neurodegenerative diseases; as a consequence, reaching higher levels of age prediction accuracy is of paramount importance to develop diagnostic tools. In this work, we propose a novel complex network model for brain based on segmenting T1-weighted MRI scans in rectangular boxes, called patches, and measuring pairwise similarities using Pearson's correlation to define a subject-specific network. We fed a deep neural network with nodal metrics, evaluating both the intensity and the uniformity of connections, to predict subjects' ages. Our model reaches high accuracies which compare favorably with state-of-the-art approaches. We observe that the complex relationships involved in this brain description cannot be accurately modeled with standard machine learning approaches, such as Ridge and Lasso regression, Random Forest, and Support Vector Machines, instead a deep neural network has to be used.

Non-Negative Matrix Factorizations for Multiplex Network Analysis

Networks have been a general tool for representing, analyzing, and modeling relational data arising in several domains. One of the most important aspect of network analysis is community detection or network clustering. Until recently, the major focus have been on discovering community structure in single (i.e., monoplex) networks. However, with the advent of relational data with multiple modalities, multiplex networks, i.e., networks composed of multiple layers representing different aspects of relations, have emerged. Consequently, community detection in multiplex network, i.e., detecting clusters of nodes shared by all layers, has become a new challenge. In this paper, we propose Network Fusion for Composite Community Extraction (NF-CCE), a new class of algorithms, based on four different non-negative matrix factorization models, capable of extracting composite communities in multiplex networks. Each algorithm works in two steps: first, it finds a non-negative, low-dimensional feature representation of each network layer; then, it fuses the feature representation of layers into a common non-negative, low-dimensional feature representation via collective factorization. The composite clusters are extracted from the common feature representation. We demonstrate the superior performance of our algorithms over the state-of-the-art methods on various types of multiplex networks, including biological, social, economic, citation, phone communication, and brain multiplex networks.

The use of multilayer network analysis in animal behaviour

Network analysis has driven key developments in research on animal behaviour by providing quantitative methods to study the social structures of animal groups and populations. A recent formalism, known as multilayer network analysis, has advanced the study of multifaceted networked systems in many disciplines. It offers novel ways to study and quantify animal behaviour through connected 'layers' of interactions. In this article, we review common questions in animal behaviour that can be studied using a multilayer approach, and we link these questions to specific analyses. We outline the types of behavioural data and questions that may be suitable to study using multilayer network analysis. We detail several multilayer methods, which can provide new insights into questions about animal sociality at individual, group, population and evolutionary levels of organization. We give examples for how to implement multilayer methods to demonstrate how taking a multilayer approach can alter inferences about social structure and the positions of individuals within such a structure. Finally, we discuss caveats to undertaking multilayer network analysis in the study of animal social networks, and we call attention to methodological challenges for the application of these approaches. Our aim is to instigate the study of new questions about animal sociality using the new toolbox of multilayer network analysis.

Emergence of synchronization in multiplex networks of mobile Rössler oscillators

Different aspects of synchronization emerging in networks of coupled oscillators have been examined prominently in the last decades. Nevertheless, little attention has been paid on the emergence of this imperative collective phenomenon in networks displaying temporal changes in the connectivity patterns. However, there are numerous practical examples where interactions are present only at certain points of time owing to physical proximity. In this work, we concentrate on exploring the emergence of interlayer and intralayer synchronization states in a multiplex dynamical network comprising of layers having mobile nodes performing two-dimensional lattice random walk. We thoroughly illustrate the impacts of the network parameters, in particular, the vision range ϕ and the step size u together with the inter- and intralayer coupling strengths ε and k on these synchronous states arising in coupled Rössler systems. The presented numerical results are very well validated by analytically derived necessary conditions for the emergence and stability of the synchronous states. Furthermore, the robustness of the states of synchrony is studied under both structural and dynamical perturbations. We find interesting results on interlayer synchronization for a continuous removal of the interlayer links as well as for progressively created static nodes. We demonstrate that the mobility parameters responsible for intralayer movement of the nodes can retrieve interlayer synchrony under such structural perturbations. For further analysis of survivability of interlayer synchrony against dynamical perturbations, we proceed through the investigation of single-node basin stability, where again the intralayer mobility properties have noticeable impacts. We also discuss the scenarios related mainly to effects of the mobility parameters in cases of varying lattice size and percolation of the whole network.

Big Data: From Forecasting to Mesoscopic Understanding. Meta-Profiling as Complex Systems

We consider Big Data as a phenomenon with acquired properties, similar to collective behaviours, that establishes virtual collective beings. We consider the occurrence of ongoing non-equivalent multiple properties in the conceptual framework of structural dynamics given by sequences of structures and not only by different values assumed by the same structure. We consider the difference between modelling and profiling in a constructivist way, as De Finetti intended probability to exist, depending on the configuration taken into consideration. The past has little or no influence, while events and their configurations are not memorised. Any configuration of events is new, and the probabilistic values to be considered are reset. As for collective behaviours, we introduce methodological and conceptual proposals using mesoscopic variables and their property profiles and meta-profile Big Data and non-computable profiles which were inspired by the use of natural computing to deal with cyber-ecosystems. The focus is on ongoing profiles, in which the arising properties trace trajectories, rather than assuming that we can foresee them based on the past.

Conserved Disease Modules Extracted From Multilayer Heterogeneous Disease and Gene Networks for Understanding Disease Mechanisms and Predicting Disease Treatments

Disease relationship studies for understanding the pathogenesis of complex diseases, diagnosis, prognosis, and drug development are important. Traditional approaches consider one type of disease data or aggregating multiple types of disease data into a single network, which results in important temporal- or context-related information loss and may distort the actual organization. Therefore, it is necessary to apply multilayer network model to consider multiple types of relationships between diseases and the important interplays between different relationships. Further, modules extracted from multilayer networks are smaller and have more overlap that better capture the actual organization. Here, we constructed a weighted four-layer disease-disease similarity network to characterize the associations at different levels between diseases. Then, a tensor-based computational framework was used to extract Conserved Disease Modules (CDMs) from the four-layer disease network. After filtering, nine significant CDMs were reserved. The statistical significance test proved the significance of the nine CDMs. Comparing with modules got from four single layer networks, CMDs are smaller, better represent the actual relationships, and contain potential disease-disease relationships. KEGG pathways enrichment analysis and literature mining further contributed to confirm that these CDMs are highly reliable. Furthermore, the CDMs can be applied to predict potential drugs for diseases. The molecular docking techniques were used to provide the direct evidence for drugs to treat related disease. Taking Rheumatoid Arthritis (RA) as a case, we found its three potential drugs Carvedilol, Metoprolol, and Ramipril. And many studies have pointed out that Carvedilol and Ramipril have an effect on RA. Overall, the CMDs extracted from multilayer networks provide us with an impressive understanding disease mechanisms from the perspective of multi-layer network and also provide an effective way to predict potential drugs for diseases based on its neighbors in a same CDM.

Solitary states in multiplex networks owing to competing interactions

Recent researches in network science demonstrate the coexistence of different types of interactions among the individuals within the same system. A wide range of situations appear in ecological and neuronal systems that incorporate positive and negative interactions. Also, there are numerous examples of systems that are best represented by the multiplex configuration. The present article investigates a possible scenario for the emergence of a newly observed remarkable phenomenon named as solitary state in coupled dynamical units in which one or a few units split off and behave differently from the other units. For this, we consider dynamical systems connected through a multiplex architecture in the presence of both positive and negative couplings. We explore our findings through analysis of the paradigmatic FitzHugh-Nagumo system in both equilibrium and periodic regimes on the top of a multiplex network having positive inter-layer and negative intra-layer interactions. We further substantiate our proposition using a periodic Lorenz system with the same scheme and show that an opposite scheme of competitive interactions may also work for the Lorenz system in the chaotic regime.

Pareto Optimality in Multilayer Network Growth

We model the formation of multilayer transportation networks as a multiobjective optimization process, where service providers compete for passengers, and the creation of routes is determined by a multiobjective cost function encoding a trade-off between efficiency and competition. The resulting model reproduces well real-world systems as diverse as airplane, train, and bus networks, thus suggesting that such systems are indeed compatible with the proposed local optimization mechanisms. In the specific case of airline transportation systems, we show that the networks of routes operated by each company are placed very close to the theoretical Pareto front in the efficiency-competition plane, and that most of the largest carriers of a continent belong to the corresponding Pareto front. Our results shed light on the fundamental role played by multiobjective optimization principles in shaping the structure of large-scale multilayer transportation systems, and provide novel insights to service providers on the strategies for the smart selection of novel routes.

Chimera states in neuronal networks: A review

Neuronal networks, similar to many other complex systems, self-organize into fascinating emergent states that are not only visually compelling, but also vital for the proper functioning of the brain. Synchronous spatiotemporal patterns, for example, play an important role in neuronal communication and plasticity, and in various cognitive processes. Recent research has shown that the coexistence of coherent and incoherent states, known as chimera states or simply chimeras, is particularly important and characteristic for neuronal systems. Chimeras have also been linked to the Parkinson's disease, epileptic seizures, and even to schizophrenia. The emergence of this unique collective behavior is due to diverse factors that characterize neuronal dynamics and the functioning of the brain in general, including neural bumps and unihemispheric slow-wave sleep in some aquatic mammals. Since their discovery, chimera states have attracted ample attention of researchers that work at the interface of physics and life sciences. We here review contemporary research dedicated to chimeras in neuronal networks, focusing on the relevance of different synaptic connections, and on the effects of different network structures and coupling setups. We also cover the emergence of different types of chimera states, we highlight their relevance in other related physical and biological systems, and we outline promising research directions for the future, including possibilities for experimental verification.

Quantifying layer similarity in multiplex networks: a systematic study

Computing layer similarities is an important way of characterizing multiplex networks because various static properties and dynamic processes depend on the relationships between layers. We provide a taxonomy and experimental evaluation of approaches to compare layers in multiplex networks. Our taxonomy includes, systematizes and extends existing approaches, and is complemented by a set of practical guidelines on how to apply them.

Finding overlapping communities in multilayer networks

Finding communities in multilayer networks is a vital step in understanding the structure and dynamics of these layers, where each layer represents a particular type of relationship between nodes in the natural world. However, most community discovery methods for multilayer networks may ignore the interplay between layers or the unique topological structure in a layer. Moreover, most of them can only detect non-overlapping communities. In this paper, we propose a new community discovery method for multilayer networks, which leverages the interplay between layers and the unique topology in a layer to reveal overlapping communities. Through a comprehensive analysis of edge behaviors within and across layers, we first calculate the similarities for edges from the same layer and the cross layers. Then, by leveraging these similarities, we can construct a dendrogram for the multilayer networks that takes both the unique topological structure and the important interplay into consideration. Finally, by introducing a new community density metric for multilayer networks, we can cut the dendrogram to get the overlapping communities for these layers. By applying our method on both synthetic and real-world datasets, we demonstrate that our method has an accurate performance in discovering overlapping communities in multilayer networks.

Null Model and Community Structure in Multiplex Networks

The multiple relationships among objects in complex systems can be described well by multiplex networks, which contain rich information of the connections between objects. The null model of networks, which can be used to quantify the specific nature of a network, is a powerful tool for analysing the structural characteristics of complex systems. However, the null model for multiplex networks remains largely unexplored. In this paper, we propose a null model for multiplex networks based on the node redundancy degree, which is a natural measure for describing the multiple relationships in multiplex networks. Based on this model, we define the modularity of multiplex networks to study the community structures in multiplex networks and demonstrate our theory in practice through community detection in four real-world networks. The results show that our model can reveal the community structures in multiplex networks and indicate that our null model is a useful approach for providing new insights into the specific nature of multiplex networks, which are difficult to quantify.

Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions

This work presents exact expressions for size distributions of weak and multilayer connected components in two generalizations of the configuration model: networks with directed edges and multiplex networks with an arbitrary number of layers. The expressions are computable in a polynomial time and, under some restrictions, are tractable from the asymptotic theory point of view. If first partial moments of the degree distribution are finite, the size distribution for two-layer connected components in multiplex networks exhibits an exponent -3/2 in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents -1/2 and -3/2.

Maximizing synchronizability of duplex networks

We study the synchronizability of duplex networks formed by two randomly generated network layers with different patterns of interlayer node connections. According to the master stability function, we use the smallest nonzero eigenvalue and the eigenratio between the largest and the second smallest eigenvalues of supra-Laplacian matrices to characterize synchronizability on various duplexes. We find that the interlayer linking weight and linking fraction have a profound impact on synchronizability of duplex networks. The increasingly large inter-layer coupling weight is found to cause either decreasing or constant synchronizability for different classes of network dynamics. In addition, negative node degree correlation across interlayer links outperforms positive degree correlation when most interlayer links are present. The reverse is true when a few interlayer links are present. The numerical results and understanding based on these representative duplex networks are illustrative and instructive for building insights into maximizing synchronizability of more realistic multiplex networks.

Multiplex recurrence networks

We have introduced a multiplex recurrence network approach by combining recurrence networks with the multiplex network approach in order to investigate multivariate time series. The potential use of this approach is demonstrated on coupled map lattices and a typical example from palaeobotany research. In both examples, topological changes in the multiplex recurrence networks allow for the detection of regime changes in their dynamics. The method goes beyond classical interpretation of pollen records by considering the vegetation as a whole and using the intrinsic similarity in the dynamics of the different regional vegetation elements. We find that the different vegetation types behave more similarly when one environmental factor acts as the dominant driving force.

Effects of temporal correlations in social multiplex networks

Multi-layered networks represent a major advance in the description of natural complex systems, and their study has shed light on new physical phenomena. Despite its importance, however, the role of the temporal dimension in their structure and function has not been investigated in much detail so far. Here we study the temporal correlations between layers exhibited by real social multiplex networks. At a basic level, the presence of such correlations implies a certain degree of predictability in the contact pattern, as we quantify by an extension of the entropy and mutual information analyses proposed for the single-layer case. At a different level, we demonstrate that temporal correlations are a signature of a 'multitasking' behavior of network agents, characterized by a higher level of switching between different social activities than expected in a uncorrelated pattern. Moreover, temporal correlations significantly affect the dynamics of coupled epidemic processes unfolding on the network. Our work opens the way for the systematic study of temporal multiplex networks and we anticipate it will be of interest to researchers in a broad array of fields.

Chimera states in a multilayer network of coupled and uncoupled neurons

We study the emergence of chimera states in a multilayer neuronal network, where one layer is composed of coupled and the other layer of uncoupled neurons. Through the multilayer structure, the layer with coupled neurons acts as the medium by means of which neurons in the uncoupled layer share information in spite of the absence of physical connections among them. Neurons in the coupled layer are connected with electrical synapses, while across the two layers, neurons are connected through chemical synapses. In both layers, the dynamics of each neuron is described by the Hindmarsh-Rose square wave bursting dynamics. We show that the presence of two different types of connecting synapses within and between the two layers, together with the multilayer network structure, plays a key role in the emergence of between-layer synchronous chimera states and patterns of synchronous clusters. In particular, we find that these chimera states can emerge in the coupled layer regardless of the range of electrical synapses. Even in all-to-all and nearest-neighbor coupling within the coupled layer, we observe qualitatively identical between-layer chimera states. Moreover, we show that the role of information transmission delay between the two layers must not be neglected, and we obtain precise parameter bounds at which chimera states can be observed. The expansion of the chimera region and annihilation of cluster and fully coherent states in the parameter plane for increasing values of inter-layer chemical synaptic time delay are illustrated using effective range measurements. These results are discussed in the light of neuronal evolution, where the coexistence of coherent and incoherent dynamics during the developmental stage is particularly likely.

Shortest Paths in Multiplex Networks

The shortest path problem is one of the most fundamental networks optimization problems. Nowadays, individuals interact in extraordinarily numerous ways through their offline and online life (e.g., co-authorship, co-workership, or retweet relation in Twitter). These interactions have two key features. First, they have a heterogeneous nature, and second, they have different strengths that are weighted based on their degree of intimacy, trustworthiness, service exchange or influence among individuals. These networks are known as multiplex networks. To our knowledge, none of the previous shortest path definitions on social interactions have properly reflected these features. In this work, we introduce a new distance measure in multiplex networks based on the concept of Pareto efficiency taking both heterogeneity and weighted nature of relations into account. We then model the problem of finding the whole set of paths as a form of multiple objective decision making and propose an exact algorithm for that. The method is evaluated on five real-world datasets to test the impact of considering weights and multiplexity in the resulting shortest paths. As an application to find the most influential nodes, we redefine the concept of betweenness centrality based on the proposed shortest paths and evaluate it on a real-world dataset from two-layer trade relation among countries between years 2000 and 2015.

A Multilayer perspective for the analysis of urban transportation systems

Public urban mobility systems are composed by several transportation modes connected together. Most studies in urban mobility and planning often ignore the multi-layer nature of transportation systems considering only aggregated versions of this complex scenario. In this work we present a model for the representation of the transportation system of an entire city as a multiplex network. Using two different perspectives, one in which each line is a layer and one in which lines of the same transportation mode are grouped together, we study the interconnected structure of 9 different cities in Europe raging from small towns to mega-cities like London and Berlin highlighting their vulnerabilities and possible improvements. Finally, for the city of Zaragoza in Spain, we also consider data about service schedule and waiting times, which allow us to create a simple yet realistic model for urban mobility able to reproduce real-world facts and to test for network improvements.

Chimera states in uncoupled neurons induced by a multilayer structure

Spatial coexistence of coherent and incoherent dynamics in network of coupled oscillators is called a chimera state. We study such chimera states in a network of neurons without any direct interactions but connected through another medium of neurons, forming a multilayer structure. The upper layer is thus made up of uncoupled neurons and the lower layer plays the role of a medium through which the neurons in the upper layer share information among each other. Hindmarsh-Rose neurons with square wave bursting dynamics are considered as nodes in both layers. In addition, we also discuss the existence of chimera states in presence of inter layer heterogeneity. The neurons in the bottom layer are globally connected through electrical synapses, while across the two layers chemical synapses are formed. According to our research, the competing effects of these two types of synapses can lead to chimera states in the upper layer of uncoupled neurons. Remarkably, we find a density-dependent threshold for the emergence of chimera states in uncoupled neurons, similar to the quorum sensing transition to a synchronized state. Finally, we examine the impact of both homogeneous and heterogeneous inter-layer information transmission delays on the observed chimera states over a wide parameter space.