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Brain network analysis can offer useful information to guide the rehabilitation of post-stroke patients. We applied functional network connection models based on multiplex-multilayer network analysis (MMN) to explore functional network connectivity changes induced by robot-aided gait training (RAGT) using the Ekso, a wearable exoskeleton, and compared it to conventional overground gait training (COGT) in chronic stroke patients. We extracted the coreness of individual nodes at multiple locations in the brain from EEG recordings obtained before and after gait training in a resting state. We found that patients provided with RAGT achieved a greater motor function recovery than those receiving COGT. This difference in clinical outcome was paralleled by greater changes in connectivity patterns among different brain areas central to motor programming and execution, as well as a recruitment of other areas beyond the sensorimotor cortices and at multiple frequency ranges, contemporarily. The magnitude of these changes correlated with motor function recovery chances. Our data suggest that the use of RAGT as an add-on treatment to COGT may provide post-stroke patients with a greater modification of the functional brain network impairment following a stroke. This might have potential clinical implications if confirmed in large clinical trials.

Cascading failure is widespread in the networks. In order to solve practical problems, it is necessary to study the cascading failure in the complex network. In this paper, we mainly focus on the cascading failure in multilayer networks with asymmetric dependence group and discuss the influence factors of networks’ robustness. We build a model with recover mechanism within the layer and define an effect between the layers. Under the effect, if the fraction of failure nodes between layers is greater than or equal to the failure–recover threshold, other connected nodes in other layers will recover and if the fraction of failure nodes between layers is less than the failure–recover threshold, the connected nodes in other layers will become invalid. It is worth mentioning that in our model, the impact of the proportion $v$ of the dependency groups and the group size $g$ on the robustness of the networks is related to the probability $p$ of initial removal. Because the failure and recovery mechanism exist, the networks robustness will be reversed by changing the value of $p$ for different values of $v$ and $g$. In addition, the number of layers $T$ has different effects on networks robustness due to the effect between the layers.

The resource allocation strategy plays an important role in the improvement of network traffic capacity. In order to solve the problem of network congestion, an efficient resource allocation strategy is proposed for multilayer networks to optimize the utilization efficiency of network resources. With the proposed strategy, the network resources are allocated to nodes more reasonable and the network congestions are evidently reduced. Simulation experiments show that the proposed resource allocation strategy can greatly improve the traffic capacity of the multilayer networks compared with the average resource allocation strategy. The proposed strategy can give full play to the traffic resources of multilayer networks, which has guiding significance for optimizing the existing networks and building new networks.

In order to study traffic dynamics on multilayer networks, it is of great significance to build a network model which can more exactly reflect the actual network layered structure characteristics. In this paper, a three-layer network model in which two logical layers are mapped on one physical layer is established, and the traffic capacities of three kinds of multilayer networks with different combinations of logical layers are compared. Simulation results show that when the physical layer is the same random network, the network whose logical layers are two random networks has the optimal traffic capacity, the network with one random network and one scale-free network in the logical layers has the better traffic capacity than the network whose logical layers are two scale-free networks.

The study of contagion dynamics in multilayer networks has become a hot topic. Networks have the characteristics of community structure. In this work, we present a comprehensive epidemic spreading model in multilayer networks with community structure. We consider the effects of population mobility, natural mortality and natural birth rate on population dynamics and contagion dynamics. The population size is determined by natural mortality and natural birth rate. The global and local epidemic thresholds are obtained. We analyze the factors which affect epidemic thresholds. The population mobility has no effect on the global epidemic threshold, but it can affect epidemic thresholds of each community. To control outbreaks of epidemics in influential communities can well control the spreading of diseases in the whole system. It is conductive to controlling outbreak of epidemics by reducing the internal average degree and interconnected links in influential communities. Numerical simulations on synthetic networks further support and enrich our conclusions.

The traffic dynamics of complex networks is closely related to network structure. By changing network structure, the traffic dynamics behavior can be optimized. Faced with the network congestion problem, we focus on the relationship between network traffic capacity and its structure. The multilayer networks are studied, which are composed of high-speed and low-speed layers. A link rewiring strategy is proposed to change the low-speed layer structure and improve the network traffic capacity. Compared with the random link rewiring strategy, the purposeful link rewiring strategy can improve network traffic capacity. A large number of simulations are carried out under the effective traffic-flow assignment strategy to prove the effectiveness of the link rewiring strategy. This strategy improves packet transmission efficiency of low-speed layer, and reduces the average length of effective path, which indicates that adjustment of low-speed layer structure can improve traffic capacity of multilayer networks.

Traffic dynamics of multilayer networks draws continuous attention from different communities since many systems are actually proved to have a multilayer structure. Since the core nodes of network are prone to congested, an effective routing strategy is of great significance to alleviate the congestion of the multilayer networks. In this paper, we propose an efficient improved routing strategy, with which the core nodes that can reasonably avoid congestion at the high-speed layer in the transmission process of packets, and can also make the most of the traffic resources of the low-speed layer nodes to optimize the traffic capacity of multilayer networks. The simulation results show that the proposed routing strategy can not only improve the network traffic capacity, but also shorten the average path length and average transmission time.

In this study, we concentrate on the synchronizability of multilayer totally homogeneous networks. The eigenvalue spectrum of four typical multilayer networks, including multilayer nearest-neighbor regular networks, multilayer small-world networks, multilayer random networks and multilayer totally homogeneous networks, has been rigorously derived and the synchronizability of four typical multilayer networks has been compared due to the master stability function. Furthermore, numerical simulations are performed to validate the synchronizability of these multilayer networks. The results indicate that multilayer totally homogeneous networks exhibit optimal synchronizability. The Euler characteristic numbers and Betti numbers of these networks are derived, and their relationships with the synchronizability of multilayer networks are analyzed. We can observe that a negative correlation is exhibited between the Euler characteristic number and the synchronizability of multilayer networks, while a positive correlation is exhibited between the Betti number and the synchronizability of multilayer networks.

During the COVID-19 pandemic, conflicting opinions on physical distancing swept across social media, affecting both human behavior and the spread of COVID-19. Inspired by such phenomena, we construct a two-layer multiplex network for the coupled spread of a disease and conflicting opinions. We model each process as a contagion. On one layer, we consider the concurrent evolution of two opinions — pro-physical-distancing and anti-physical-distancing — that compete with each other and have mutual immunity to each other. The disease evolves on the other layer, and individuals are less likely (respectively, more likely) to become infected when they adopt the pro-physical-distancing (respectively, anti-physical-distancing) opinion. We develop approximations of mean-field type by generalizing monolayer pair approximations to multilayer networks; these approximations agree well with Monte Carlo simulations for a broad range of parameters and several network structures. Through numerical simulations, we illustrate the influence of opinion dynamics on the spread of the disease from complex interactions both between the two conflicting opinions and between the opinions and the disease. We find that lengthening the duration that individuals hold an opinion may help suppress disease transmission, and we demonstrate that increasing the cross-layer correlations or intra-layer correlations of node degrees may lead to fewer individuals becoming infected with the disease.

The controllability of multilayer networks has become increasingly important in many areas of science and engineering. In this paper, we identify the general rules that determine the controllability and control energy cost of multilayer networks. First, we quantitatively estimate the control energy cost of multilayer networks and investigate the impacts of different coupling strength and coupling patterns on the control energy cost for multilayer networks. The results indicate that the average energy and the coupling strength have an approximately linear relationship in multilayer networks with two layers. Second, we study how the coupling strength and the connection patterns between different layers affect the controllability of multilayer networks from both theoretical and numerical aspects. The obtained piecewise functional relations between the controllability’s measure and coupling strength reveal the existence of an optimal coupling strength for the different interconnection strategies in multilayer networks. In particular, the numerical experiments demonstrate that there exists a tradeoff between the optimal controllability and optimal control energy for selecting interlayer connection patterns in multilayer networks. These results provide a comprehensive understanding of the impact of interlayer couplings on the controllability and control energy cost for multilayer networks and provide a methodology for selecting the control nodes and coupling strength to maximize the controllability and minimize the control energy cost.