The introduction of a bridge module between multiple networks may bring forth new possibility of collective dynamical behaviors. In this paper, by exploring the sustained oscillatory behaviors of bridge-module-connected excitable networks, a double-mode oscillation can self-organize to emerge besides the single-mode oscillation mode, where one network performs a fast-wave oscillation while the other implements a slow-wave oscillation. This behavior is similar to the unihemispheric sleep phenomenon widely observed in birds and aquatic mammals, and is consequently named as the unihemispheric-sleep-like oscillation (USLO). By utilizing the dominant phase-advanced driving method, the loop structure is demonstrated as the oscillation source in maintaining the USLOs. More profoundly, an effective-driving analysis approach is proposed to unveil the mechanism of the USLOs. The bridge-module-induced instability of the original 1:1 response mode is shown as the determinant in inducing the USLOs on the bridge-module-connected excitable networks. Furthermore, the bridge-module-induced USLOs on multiple connected excitable networks are found to be universal. The method and results demonstrated in the present paper can give us clues in understanding the amazing phenomenon of unihemispheric sleep emerging in highly complex and heterogeneous brain network systems.

Complex dynamical systems often exhibit distinct behaviors under varying control parameters and environmental conditions, posing a major challenge for unified modeling and prediction. Conventional reservoir computing (RC) is typically restricted to fixed-parameter settings and therefore has limited ability to capture parameter-dependent dynamical evolution. To address this limitation, we propose a parameter-aware reservoir computing framework (PARC) for learning parameterized nonlinear dynamical systems with delay and memory effects. By introducing an additional parameter channel, PARC jointly embeds control parameters and time series into a unified high-dimensional reservoir space. In addition, delays are incorporated into the hyperparameter optimization scheme, enabling the model to better adapt to delay-induced dynamics. We validate PARC on delayed logistic systems under both noiseless and noisy observation settings. The results show that PARC achieves stable long-horizon prediction and accurately reconstructs the attractor geometry and statistical “climate” of the underlying dynamics. Generalization experiments under unseen parameter conditions further demonstrate that the proposed framework learns a parameter-dependent family of dynamical evolutions rather than a single trajectory mapping. We further apply PARC to NDVI time series as a real-world example of a spatiotemporal dynamical system with delay and heterogeneity. These results indicate that PARC provides a unified and robust framework for learning and predicting parameterized nonlinear dynamical systems.

Television remains an indispensable medium for information and entertainment, even in the era of widespread streaming media. With the expansion of TV channels through set-top boxes, users now face an overwhelming variety of choices, leading to information overload problems. Recommendation systems have effectively solved the information overload problem and can thus be naturally applied to television. Prior research has focused on improvements in algorithms and the addition of other data. In this paper, without introducing external data, we generate recommendations based on 3 months of TV viewing data from a Chinese city. Considering the large amount of noisy data caused by short stays in TV programs, we simplify the original almost fully connected tripartite network by eliminating the insignificant links with the Revealed Comparative Advantage (RCA) metric to comprehensively reflect user preferences. The inclusion of channel nodes allows the network structure to better align with user behavior characteristics, which differs from traditional bipartite networks that only include user-program interactions. We examine data with different sparsity and find that our approach continues to outperform conventional bipartite network recommendations in terms of accuracy. The advantages of our approach have been validated through comparisons with other advanced methods and across different datasets. Overall, only based on viewing records of users, our work provides accurate TV program recommendations that can capture the underlying user behavior characteristics.

We investigated synchronization behavior using an experimental setup consisting of two metronomes placed on a platform floating over water. By setting the metronomes to oscillate perpendicular to the line between them, we observed three distinct modes of movement: in-phase synchronization, anti-phase synchronization, and synchronization with a fixed phase difference. While this last mode resembles phase-locking, it is important to distinguish that phase-locking typically refers to an oscillator’s response to external pacing, whereas the fixed phase difference observed in our study emerges from the mutual interaction between two metronomes. The frequencies of oscillations, and the placement of the metronomes are also changed to check the reliability of the new phenomenon. Even if we changed the material of the platform to a heavier one or turned around one of the metronomes, synchronization with a fixed time delay still was still observed. Drawing on previous research, we developed mathematical equations to model the coupled metronomes and performed numerical simulations that successfully reproduced all three observed phenomena. The simulation results showed excellent agreement with our experimental observations. These findings contribute to our understanding of coupled oscillators and may have potential applications in various fields.

Spatial networks play a central role in the analysis of real-world systems' structures. However, the modelling of spatial networks remains a challenge, primarily due to the elusive mechanisms governing the emergence of loops. Since loops are intimately related to network functionality, it is imperative to gain an understanding of their formation. Early models, often based on top-down planning, tend to overlook the evolving nature of spatial networks, resulting in exceedingly high computational complexity. To address these limitations, self-organized models have been introduced, though they typically yield tree structures or are designed for specific types of networks. In this paper, to facilitate the classification and reconstruction of spatial networks, we propose a class of economical-efficient models. They consider cost and efficiency as the primary driving forces behind network evolution. Different network patterns emerge as a consequence of the interplay between node heterogeneity and spatial constraints. Our model is grounded in the concept of self-organization and demonstrates the ability to replicate many macroscopic properties observed in real-world systems. The model can also serve as a tool for network reconstruction as we further show how to fit the parameter and apply it to domestic airline networks.

Protecting target nodes in complex networks is a critical issue in network security research. In many real-world scenarios, the identities of certain target nodes remain unknown, and the impact of this incomplete information on appropriately selecting initial spreaders is not fully understood. This paper first examines how the observability rate of target nodes affects the effectiveness of targeted spreading. The findings indicate that even if most target nodes are unobservable, identifying the optimal spreader for targeted propagation is still feasible in many real-world networks. This indicates that solely relying on protecting target nodes through external observation avoidance is insufficient. To address this issue, we developed a novel camouflage defense strategy for target nodes in complex networks by integrating target centrality, the distribution of target node groups, and the network distance between disguised and hidden target nodes. This strategy effectively hinders attackers’ selection of the optimal initial spreader by adjusting the visibility of selected target nodes and their neighbors, without altering the network structure. Finally, we validate the effectiveness of the proposed model in three aspects: matching accuracy of the optimal initial spreader, implementation of SIR propagation dynamics, and comparative testing against other models. These results were obtained not only from three types of generic artificial networks but also from multiple real datasets.

Identifying spiral wave tips of spatiotemporal dynamical systems from time series represents a significant challenge for understanding and controlling complex dynamics. Many previous methods for calculating tips relied on phase analysis, and they inevitably needed to set a phase origin and required multiple time slices for phase calculation. Reservoir computing, a simplified recurrent neural network paradigm, has spurred many investigations in several fields to capture and predict the features of complex, nonlinear dynamics. Based on the superior performance of reservoir computing, we investigated its application in analyzing spiral wave tips in reaction–diffusion systems. In this paper, we employ reservoir computing to identify spiral wave tips in some simple cases (spiral waves modeled by CGLE with one or two tips) and demonstrated that our model could accurately identify tips using only one time slice. Furthermore, we confirmed that the model maintained high accuracy in identifying tips of moving one-tip spiral waves in other systems (Bär, FHN). Moreover, we analyzed complex cases (evolving spiral waves and turbulence), with results indicating effective model performance. Ultimately, we demonstrated reservoir computing’s robustness, noting its superior performance over conventional algorithms when handling data contaminated with noise from the sampling process. In summary, reservoir computing exhibits low computational complexity, requires minimal data and fewer constraints, and achieves high accuracy. This approach offers novel prospects for identifying topological structures in practical applications, such as monitoring and controlling spiral wave tips in cardiac illnesses.

The study of specific physiological processes from the perspective of network physiology has gained recent attention. Modeling the global information integration among the separated functionalized modules in structural and functional brain networks is a central problem. In this article, the preferentially cutting–rewiring operation (PCRO) is introduced to approximatively describe the above physiological process, which consists of the cutting procedure and the rewiring procedure with specific preferential constraints. By applying the PCRO on the classical Erdös–Rényi random network (ERRN), three types of isolated nodes are generated, based on which the common leaves (CLs) are formed between the two hubs. This makes the initially homogeneous ERRN experience drastic changes and become heterogeneous. Importantly, a statistical analysis method is proposed to theoretically analyze the statistical properties of an ERRN with a PCRO. Specifically, the probability distributions of these three types of isolated nodes are derived, based on which the probability distribution of the CLs can be obtained easily. Furthermore, the validity and universality of our statistical analysis method have been confirmed in numerical experiments. Our contributions may shed light on a new perspective in the interdisciplinary field of complexity science and biological science and would be of great and general interest to network physiology.

Spontaneous activity of the human brain provides a window to explore intrinsic principles of functional organization. However, most studies have focused on interregional functional connectivity. The principles underlying rich repertoires of instantaneous activity remain largely unknown. We apply a recently proposed eigen-microstate analysis to three resting-state functional MRI datasets to identify basic modes that represent fundamental activity patterns that coexist over time. We identify five leading basic modes that dominate activity fluctuations. Each mode exhibits a distinct functional system-dependent coactivation pattern and corresponds to specific cognitive profiles. In particular, the spatial pattern of the first leading basis mode shows the separation of activity between the default-mode and primary and attention regions. Based on theoretical modelling, we further reconstruct individual functional connectivity as the weighted superposition of coactivation patterns corresponding to these leading basic modes. Moreover, these leading basic modes capture sleep deprivation-induced changes in brain activity and interregional connectivity, primarily involving the default-mode and task-positive regions. Our findings reveal a dominant set of basic modes of spontaneous activity that reflect multiplexed interregional coordination and drive conventional functional connectivity, furthering the understanding of the functional significance of spontaneous brain activity.

People are increasingly involved in online activities. Online activities can be regarded as movements in virtual space, such as jumping from webpage to webpage while surfing online, switching channels while watching TV, and browsing commodities while shopping online, which can affect information propagation, innovation spread, social activities, etc. Most previous efforts have been devoted to modeling the scaling properties of human mobility in physical space. Few studies aim to establish a unified and integral model to understand the fundamental dynamics underlying human virtual mobility. In this paper, we study human mobility in virtual space empirically and theoretically based on two datasets involving TV watching and online shopping and attempt to answer three unsolved issues. First, human virtual mobility shares common features, supported by the fact that striking agreements appear in the scaling properties of both datasets. Second, there exists a universal rule governing an individual's choice in virtual mobility, which is distinct from that in the real world due to travel restrictions. Third, there exists a unified model incorporating the behavior rule unique to virtual space under the framework of Exploration and Preferential Return, which can be used to reproduce the scaling properties of virtual mobility. We reveal the mechanism behind human virtual mobility through consistent scaling properties and develop a corresponding dynamic model based on empirical data.

Television is the primary medium through which most families access entertainment and information in their daily lives. Thus, understanding users’ TV viewing behavior is meaningful for several practical issues, such as evaluating the influence of TV channels and providing personalized TV recommendations. However, most existing works regarding TV viewing data are limited to basic statistics (e.g., TV ratings). In this paper, we analyze a large-scale TV viewing dataset for a city in China via a complex network approach. We construct a directed network that characterizes the collective channel-switching behavior of viewers. By using the PageRank method, we reveal the influential TV channels that are more in line with people’s expectations than their rankings based on simple TV ratings. We further construct a network in which channels are linked according to their similarity in users’ switching preferences. This network exhibits a clear community structure, which can help TV stations understand which channels are in the bottleneck and which channels have potential. Overall, our work provides a system perspective to evaluate TV channels and their relationships.

Reservoir computing, a new method of machine learning, has recently been used to predict the state evolution of various chaotic dynamic systems. It has significant advantages in terms of training cost and adjusted parameters; however, the prediction length is limited. For classic reservoir computing, the prediction length can only reach five to six Lyapunov times. Here, we modified the method of reservoir computing by adding feedback, continuous or discrete, to "calibrate" the input of the reservoir and then reconstruct the entire dynamic systems. The reconstruction length appreciably increased and the training length obviously decreased. The reconstructing of dynamical systems is studied in detail under this method. The reconstruction can be significantly improved both in length and accuracy. Additionally, we summarized the effect of different kinds of input feedback. The more it interacts with others in dynamical equations, the better the reconstructions. Nonlinear terms can reveal more information than linear terms once the interaction terms are equal. This method has proven effective via several classical chaotic systems. It can be superior to traditional reservoir computing in reconstruction, provides new hints in computing promotion, and may be used in some real applications.

Extremely large-scale networks have received increasing attention in recent years. The development of big data and network science provides an unprecedented opportunity for research on these networks. However, it is difficult to perform analysis directly on numerous real networks due to their large size. A solution is to sample a subnetwork instead for detailed research. Unfortunately, the properties of the subnetworks could be substantially different from those of the original networks. In this context, a comprehensive understanding of the sampling methods would be crucial for network-based big data analysis. In our work, we find that the sampling deviation is the collective effect of both the network heterogeneity and the biases caused by the sampling methods themselves. Here, we study the widely used random node sampling (RNS), breadth-first search, and a hybrid method that falls between these two. We empirically and analytically investigate the differences in topological properties between the sampled network and the original network under these sampling methods. Empirically, the hybrid method has the advantage of preserving structural properties in most cases, which suggests that this method performs better with no additional information needed. However, not all the biases caused by sampling methods follow the same pattern. For instance, properties, such as link density, are better preserved by RNS. Finally, models are constructed to explain the biases concerning the size of giant connected components and link density analytically.

Network diffusion processes play an important role in solving the information overload problem. It has been shown that the diffusion-based recommendation methods have the advantage to generate both accurate and diverse recommendation items for online users. Despite that, numerous existing works consider the rating information as link weight or threshold to retain the useful links, few studies use the rating information to evaluate the recommendation results. In this paper, we measure the average rating of the recommended products, finding that diffusion-based recommendation methods have the risk of recommending low-rated products to users. In addition, we use the rating information to improve the network-based recommendation algorithms. The idea is to aggregate the diffusion results on multiple user-item bipartite networks each of which contains only links of certain ratings. By tuning the parameters, we find that the new method can sacrifice slightly the recommendation accuracy for improving the average rating of the recommended products.

In this paper, we extensively investigate the oscillation-mode dynamics in excitable homogeneous random networks (EHRNs) and the corresponding determinants. It is exposed that, at a certain set of system parameters, there exists two kinds of oscillation modes, i.e., the major oscillation mode (MajOM) and the minor oscillation mode (MinOM). Moreover, the dynamics of these two mode groups can be effectively impacted by the system parameters, and two types of oscillation-mode dynamics, i.e., transfer and transition, have been revealed explicitly. Importantly, the minimum Winfree loop (MWL) and the excitation threshold have been exposed as the determinants of oscillation-mode dynamics in EHRNs, and can respectively induce the oscillation-mode transfer and transition. We do hope our results will shed light on a deeper understanding of oscillation-mode dynamics and corresponding determinants in some specific physiological processes of actual biological systems and will have a useful impact in related fields.

Various behaviours of nonlinear wave propagation and competition have been discussed and investigated extensively and meticulously, especially when the media are homogeneous. However, corresponding studies in heterogeneous media are much scarcer. In this paper, spontaneously generated waves from one-dimensional heterogeneous oscillatory media, modelled by complex Ginzburg–Landau equations with spatially varied controlling parameters, are investigated. An unexpected homogeneous wave train clearly emerges under certain conditions. With the theory of interface-selected waves, we can theoretically predict the frequencies and wavenumbers under several conditions. This kind of wave train can be found in a wide region of parameter space. These phenomena are robust when parameters are varied nonlinearly or linearly with fluctuation. Moreover, this kind of homogeneous wave plays an important role in wave competition and affects wave propagation in spatially heterogeneous nonlinear systems, which will bring new applications of heterogeneity and provide new ideas for wave control.

The effects of spatial heterogeneity on a two-dimensional complex Ginzburg–Landau equation model are studied. In general, the interaction of a pair of spiral waves with a large degree of heterogeneity in two different media will cause three different patterns: (a) Multiple spiral waves coexist in different media; (b) the spiral wave is swept away in one medium and remains in another medium; (c) all of the spiral waves are suppressed by travelling waves having different frequencies. These travelling waves are generated from interface reported before. It is found that the interface is a wave source that can generate travelling waves with different frequencies in two submedia to compete with the original spiral waves in two different media. The competition results depend on the frequencies of the original spiral wave and the two travelling waves. Furthermore, local periodic pacing can replace the effect of the interface and reproduce the corresponding results, which gives additional evidence that the interface works as a wave source. The results give new ideas in pattern control such that we can suppress and annihilate spiral waves by generating a large degree of heterogeneity using selected parameters.

Synchronization rhythm and oscillating in biological systems can give clues to understanding the cooperation and competition between cells under appropriate biological and physical conditions. As a result, the network setting is appreciated to detect the stability and transition of collective behaviors in a network with different connection types. In this paper, the synchronization performance in time-delayed excitable homogeneous random networks (EHRNs) induced by diversity in system parameters is investigated by calculating the synchronization parameter and plotting the spatiotemporal evolution pattern, and distinct impacts induced by parameter-diversity are detected by setting different time delays. It is found that diversity has no distinct effect on the synchronization performance in EHRNs with small time delay being considered. When time delay is increased greatly, the synchronization performance of EHRN degenerates remarkably as diversity is increased. Surprisingly, by setting a moderate time delay, appropriate parameter-diversity can promote the synchronization performance in EHRNs, and can induce the synchronization transition from the asynchronous state to the weak synchronization. Moreover, the bistability phenomenon, which contains the states of asynchronous state and weak synchronization, is observed. Particularly, it is confirmed that the parameter-diversity promoted synchronization performance in time-delayed EHRN is manifested in the enhancement of the synchronization performance of individual oscillation and the increase of the number of synchronization transitions from the asynchronous state to the weak synchronization. Finally, we have revealed that this kind of parameter-diversity promoted synchronization performance is a robust phenomenon.

The investigation of self-sustained oscillations in excitable complex networks is very important in understanding various activities in brain systems, among which the exploration of the key determinants of oscillations is a challenging task. In this paper, by investigating the influence of system parameters on self-sustained oscillations in excitable Erdös-Rényi random networks (EERRNs), the minimum Winfree loop (MWL) is revealed to be the key factor in determining the emergence of collective oscillations. Specifically, the one-to-one correspondence between the optimal connection probability (OCP) and the MWL length is exposed. Moreover, many important quantities such as the lower critical connection probability (LCCP), the OCP, and the upper critical connection probability (UCCP) are determined by the MWL. Most importantly, they can be approximately predicted by the network structure analysis, which have been verified in numerical simulations. Our results will be of great importance to help us in understanding the key factors in determining persistent activities in biological systems.

Wave propagation is an important characteristic for pattern formation and pattern dynamics. To date, various waves in homogeneous media have been investigated extensively and have been understood to a great extent. However, the wave behaviors in heterogeneous media have been studied and understood much less. In this work, we investigate waves that are spontaneously generated in one-dimensional heterogeneous oscillatory media governed by complex Ginzburg-Landau equations; the heterogeneity is modeled by multiple interacting homogeneous media with different system control parameters. Rich behaviors can be observed by varying the control parameters of the systems, whereas the behavior is incomparably simple in the homogeneous cases. These diverse behaviors can be fully understood and physically explained well based on three aspects: dispersion relation curves, driving-response relations, and wave competition rules in homogeneous systems. Possible applications of heterogeneity-generated waves are anticipated.