Understanding how information propagation affects epidemic dynamics has become an emerging topic of interest. However, the influence of interpersonal relationship heterogeneity on information acquisition and disease transmission has been largely overlooked. In this work, we introduce a hypergraph structure for Cyber-Physical Systems (CPSs) with two distinct layers. The upper layer, referred to as the cyber layer, consists of a mixed hypergraph, capturing both pairwise propagation and higher-order diffusion of epidemic-related information. The lower layer, referred to as the physical layer, employs a Susceptible-Infected-Susceptible (SIS) process to capture epidemic spreading. This work introduces an adaptive perception-protection mechanism based on Jaccard similarity, which accounts for interpersonal heterogeneity. In this mechanism, individuals receive information based on their relationships with neighbors and take protective measures accordingly. We analyze the impact of interpersonal relationships and the adoption of neighborhood-based self-protection strategies on epidemic dynamics. Furthermore, we conduct a theoretical analysis based on the Microscopic Markov Chain Approach (MMCA), analytically derive the outbreak threshold, and confirm the results with extensive Monte Carlo (MC) simulations. The results show that stronger interpersonal relationships can promote information propagation, significantly increase the threshold for epidemic outbreaks, and effectively suppress the scale of the epidemic. The study provides theoretical support for designing epidemic control strategies considering interpersonal heterogeneity and improves the understanding of epidemic spreading on hypergraphs.

The game interactions among individuals in nature are often uncertain and dynamically evolving, significantly influencing the persistence of cooperation. However, it remains a formidable challenge to effectively characterize these dynamic properties in structured populations, derive theoretical conditions for cooperation, and identify the optimal game distribution for promoting cooperation. To address these issues, we propose the variable game framework in a structured population, where the game interactions between different individuals change over time. By means of the Markov chain and the pair approximation method, we derive theoretical conditions under which cooperation is favored by natural selection and when it is favored over defection under weak selection. Furthermore, we, respectively, formulate and solve two optimization problems to determine the optimal game distribution that most effectively fosters the evolution of cooperation by maximizing the gradient of cooperation selection and minimizing the fitness difference between defectors and cooperators. The theoretical predictions regarding both the conditions for cooperation and optimal game distribution are further validated by numerical calculations and extensive Monte Carlo simulations. Our findings offer novel insights into the mechanisms driving cooperative behavior in complex systems and provide theoretical guidance for designing optimal game environments that facilitate the evolution of cooperation.

Understanding how human mobility and information propagation influence the course of an epidemic remains a key challenge in digital epidemiology. In this work, we develop a new awareness-based, event-triggered epidemic model embedded within a networked Cyber-Physical System (CPS). In our framework, disease transmission and the dissemination of epidemic-related information evolve together on two interconnected layers. In detail, the physical layer models dis ease spread through human movement between two types of locations–residences and transfer stations-forming a bipartite metapopulation network. This structure captures the rendezvous effect, which reflects how gatherings in shared locations contribute to infection spread. The cyber layer represents the flow of information through digital communication networks. We introduce an event-triggered migration regulation mechanism, whereby individuals adapt their movement patterns based on local awareness thresholds, leading to a decentralized control process embedded within the network. Using a microscopic Markov chain approach (MMCA), we derive the epidemic threshold analytically and validate our results through extensive Monte Carlo simulations. Our findings show that event-triggered migration effectively suppresses the overall spread of the disease and lowers infection peaks-especially in heterogeneous populations and densely connected gathering points. These results demonstrate the potential of CPS-based epidemic models to enable real-time, awareness-driven interventions and to inform the design of decentralized control strategies that leverage digital communication dynamics.

Many real-world scale-free networks, such as neural networks and online communication networks, consist of a fixed number of nodes but exhibit dynamic edge fluctuations. However, traditional models frequently overlook scenarios where the node count remains constant, instead prioritizing node growth. In this work, we depart from the assumptions of node number variation and preferential attachment to present an innovative model that conceptualizes node degree fluctuations as a state-dependent random walk process with stasis and variable diffusion coefficient. We show that this model yields stochastic dynamic networks with stable scale-free properties. Through comprehensive theoretical and numerical analyses, we demonstrate that the degree distribution converges to a power-law distribution, provided that the lowest degree state within the network is not an absorbing state. Furthermore, we investigate the resilience of the fraction of the largest component and the average shortest path length following deliberate attacks on the network. By using three real-world networks, we confirm that the proposed model accurately replicates actual data. The proposed model thus elucidates mechanisms by which networks, devoid of growth and preferential attachment features, can still exhibit power-law distributions and be used to simulate and study the resilience of attacked fixed-size scale-free networks.

Complex networks serve as abstract models for understanding real-world complex systems and provide frameworks for studying structured dynamical systems. This article addresses limitations in current studies on the exploration of individual birth-death and the development of community structures within dynamic systems. To bridge this gap, we propose a networked evolution model that includes the birth and death of individuals, incorporating reinforcement learning through games among individuals. Each individual has a lifespan following an arbitrary distribution, engages in games with network neighbors, selects actions using Q-learning in reinforcement learning, and moves within a two-dimensional space. The developed theories are validated through extensive experiments. Besides, we observe the evolution of cooperative behaviors and community structures in systems both with and without the birth-death process. The fitting of real-world populations and networks demonstrates the practicality of our model. Furthermore, comprehensive analyses of the model reveal that exploitation rates and payoff parameters determine the emergence of communities, learning rates affect the speed of community formation, discount factors influence stability, and two-dimensional space dimensions dictate community size. Our model offers a novel perspective on real-world community development and provides a valuable framework for studying population dynamics behaviors.

Since Granell et al. proposed a multiplex network for information and epidemic propagation, researchers have explored how information propagation affects epidemic dynamics. However, the role of individuals acquiring information through physical interactions has received relatively less attention. In this work, we introduce a novel source of information: physical-layer information, and derive the epidemic outbreak threshold using the Microscopic Markov Chain Approach (MMCA). Our simulation results indicate that the outbreak threshold derived from the MMCA is consistent with the Monte Carlo (MC) simulation results, thereby confirming the accuracy of the theoretical model. Furthermore, we find that the physical-layer information effectively increases the population’s awareness density and the infection threshold$\beta _{c}$, while reducing the population’s infection density, thereby suppressing the spreading of the epidemic. Another interesting finding is that when the density of 2-simplex information is relatively high, the 2-simplex plays a role similar to pairwise interaction, significantly enhancing the population’s awareness density and effectively preventing large-scale epidemic outbreaks. In addition, our model works equally well for cyber-physical systems with similar interaction mechanisms, while we simulate and validate it in a real grid system.

Complex network theory provides a unifying framework for the study of structured dynamic systems. The current literature emphasizes a widely reported phenomenon of intermittent interaction among network vertices. In this article, we introduce a complex network model that considers the stochastic switching of individuals between activated and quiescent states at power-law rates and the corresponding evolutionary dynamics. By using the Markov chain and renewal theory, we discover a homogeneous stationary distribution of activated sizes in the network with power-law activating patterns and infer some statistical characteristics. To better understand the effect of power-law activating patterns, we study the two-person-two-strategy evolutionary game dynamics, demonstrate the absorbability of strategies, and obtain the critical cooperation conditions for prisoner’s dilemmas in homogeneous networks without mutation. The evolutionary dynamics in real networks are also discussed. Our results provide a new perspective to analyze and understand social physics in time-evolving network systems.

Open data, as an essential element in the sustainable development of the digital economy, is highly valued by many relevant sectors in the implementation process. However, most studies suppose that there are only data providers and users in the open data process and ignore the existence of data regulators. In order to establish long-term green supply relationships between multistakeholders, we hereby introduce data regulators and propose an evolutionary game model to observe the cooperation tendency of multistakeholders (data providers, users, and regulators). The newly proposed game model enables us to intensively study the trading behavior which can be realized as strategies and payoff functions of the data providers, users, and regulators. Besides, a replicator dynamic system is built to study evolutionary stable strategies of multistakeholders. In simulations, we investigate the evolution of the cooperation ratio as time progresses under different parameters, which is proved to be in agreement with our theoretical analysis. Furthermore, we explore the influence of the cost of data users to acquire data, the value of open data, the reward (penalty) from the regulators, and the data mining capability of data users to group strategies and uncover some regular patterns. Some meaningful results are also obtained through simulations, which can guide stakeholders to make better decisions in the future.

The psychology of the individual is continuously changing in nature, which has a significant influence on the evolutionary dynamics of populations. To study the influence of the continuously changing psychology of individuals on the behavior of populations, in this article, we consider the game transitions of individuals in evolutionary processes to capture the changing psychology of individuals in reality, where the game that individuals will play shifts as time progresses and is related to the transition rates between different games. Besides, the individual’s reputation is taken into account and utilized to choose a suitable neighbor for the strategy updating of the individual. Within this model, we investigate the statistical number of individuals staying in different game states and the expected number fits well with our theoretical results. Furthermore, we explore the impact of transition rates between different game states, payoff parameters, the reputation mechanism, and different time scales of strategy updates on cooperative behavior, and our findings demonstrate that both the transition rates and reputation mechanism have a remarkable influence on the evolution of cooperation. Additionally, we examine the relationship between network size and cooperation frequency, providing valuable insights into the robustness of the model.

Dynamic processes in complex network are crucial for better understanding collective behavior in human societies, biological systems, and the Internet. In this article, we first focus on the continuous Markov-based modeling of evolving networks with the birth-death of individuals. A new individual arrives at the group by the Poisson process, while new links are established in the network through either uniform connection or preferential attachment. Moreover, an existing individual has a limited lifespan before leaving the network. We determine stationary topological properties of these networks, including their size and mean degree. To address the effect of the birth-death evolution, we further study the information dynamics in the proposed network model from the random drift and natural selection perspective, based on assumptions of total-stochastic and fitness-driven evolution, respectively. In simulations, we analyze the fixation probability of individual information and find that means of new connections affect the random drift process but do not affect the natural selection process.

Ever since the Barabási–Albert (BA) scale-free network has been proposed, network modeling has been studied intensively in light of the network growth and the preferential attachment (PA). However, numerous real systems are featured with a dynamic evolution including network reduction in addition to network growth. In this article, we propose a novel mechanism for evolving networks from the perspective of vertex degree. We construct a queueing system to describe the increase and decrease of vertex degree, which drives the network evolution. In our mechanism, the degree increase rate is regarded as a function positively correlated to the degree of a vertex, ensuring the PA in a new way. Degree distributions are investigated under two expressions of the degree increase rate, one of which manifests a “long tail,” and another one varies with different values of parameters. In simulations, we compare our theoretical distributions with simulation results and also apply them to real networks, which presents the validity and applicability of our model.

With the prevalence of COVID-19, the modeling of epidemic propagation and its analyses have played a significant role in controlling epidemics. However, individual behaviors, in particular the self-protection and migration, which have a strong influence on epidemic propagation, were always neglected in previous studies. In this paper, we mainly propose two models from the individual and population perspectives. In the first individual model, we introduce the individual protection degree that effectively suppresses the epidemic level as a stochastic variable to the SIRS model. In the alternative population model, an open Markov queueing network is constructed to investigate the individual number of each epidemic state, and we present an evolving population network via the migration of people. Besides, stochastic methods are applied to analyze both models. In various simulations, the infected probability, the number of individuals in each state and its limited distribution are demonstrated.

Evolving networks have always been studied a lot featuring the dynamic properties of real-life networks. Studying the mechanism of the growth and death of a network is of great significance to network modeling. Identical to many models focused on the growing process, in this article, we study the decreasing process thoroughly. A novel evolving network model considering the growing and decreasing process is established based on the queueing system. Focused on the degreasing process, we originally investigate two strategies of vertex deleting that are the brutal strategy and the heritable strategy which characterizes the heritable behavior of “dying” vertices in real networks. On the basis of our model, stochastic properties of the proposed network are analyzed, e.g., the distribution and the expectation of the stationary scale of the network are theoretically obtained. In addition to that, degree distributions with different strategies are demonstrated in simulations, which manifests the power-low distribution. The reliability of the network is also studied by different attacks, sharing the same characteristic with the scale-free network.

In the Internet of Things (IoT), asynchronous networks with varying topology are quite common. Meanwhile, Gaussian noise and impulsive noise widely exist in asynchronous networks. Existing works on distributed estimation problems in networks primarily consider fixed topologies and Gaussian noise. Thus, these algorithms are not suitable for distributed parameter estimation in asynchronous networks. To overcome this issue, we propose a distributed diffusion kernel risk-sensitive loss (d-KRSL) algorithm, which can achieve a good performance in asynchronous networks with varying topology, and maintains the robustness to both Gaussian and impulsive noise. The mean and mean square performances of the proposed algorithm are analyzed theoretically and verified by numerical simulation results.

During the past decades, power-law distributions have played a significant role in analyzing the topology of scale-free networks. However, in the observation of degree distributions in practical networks and other nonuniform distributions such as the wealth distribution, we discover that, there exists a peak at the beginning of most real distributions, which cannot be accurately described by a monotonic decreasing power-law distribution. To better describe the real distributions, in this paper, we propose a subnormal distribution derived from evolving networks with variable elements and study its statistical properties for the first time. By utilizing this distribution, we can precisely describe those distributions commonly existing in the real world, e.g., distributions of degree in social networks and personal wealth. Additionally, we fit connectivity in evolving networks and the data observed in the real world by the proposed subnormal distribution, resulting in a better performance of fitness.

Since the great mathematician Leonhard Euler initiated the study of graph theory, the network has been one of the most significant research subject in multidisciplinary. In recent years, the proposition of the small-world and scale-free properties of complex networks in statistical physics made the network science intriguing again for many researchers. One of the challenges of the network science is to propose rational models for complex networks. In this paper, in order to reveal the influence of the vertex generating mechanism of complex networks, we propose three novel models based on the homogeneous Poisson, nonhomogeneous Poisson and birth death process, respectively, which can be regarded as typical scale-free networks and utilized to simulate practical networks. The degree distribution and exponent are analyzed and explained in mathematics by different approaches. In the simulation, we display the modeling process, the degree distribution of empirical data by statistical methods, and reliability of proposed networks, results show our models follow the features of typical complex networks. Finally, some future challenges for complex systems are discussed.

With the rapid development of science technology and explosively increasing of network's complication, complex network as an emerging research hotspot has attracted more and more scientists' attention. Meanwhile, search strategy as a main research method in complex network plays a more and more important role on study of complex network, and it has great practical significance and research value. So many classical search algorithms have been proposed according to network's specialty, such as breadth-first search (BFS), random walk (RW), and high degree seeking (HDS). Unfortunately, a flawless solution for all kinds of models in complex network has not been presented so far due to the above algorithms are only suitable for some special circumstances. For improving the search efficiency, this paper appears an improved strategy which has a hybrid merit combining both high efficiency and low consumption. This strategy augments a compressed process to save useful path's information, so we can use the stored data in the search process to effectively reduce search step and query flow. In the simulation, the proposed algorithm will be compared with HDS in the models of complex network which have diverse type or different size. The result of simulation was used to illustrate the efficient performance of this strategy and demonstrate that the proposed search algorithm can produce a better fruit than others.