Advances in Complex Systems

The continuous increase in unemployment rates and their significant economic impact necessitate the rapid updating and modification of present models and policies implemented by governmental entities. To successfully handle the timely transmission of employment within the workforce, many contemporary models still need the incorporation of an individual’s job history. Consequently, in order to study the unemployment problem, this research presents a multi-order fractional nonlinear mathematical model that takes into account the Caputo fractional order derivative and three important variables: the number of skilled unemployed individuals, the number of employed individuals, and the number of open positions. The existence and uniqueness of the proposed model’s solution are demonstrated by using generalization of Picard fixed point theorem. The solution of the proposed model is bounded and non-negative. The reproduction number has been analyzed to determine the factors that would help create new job vacancies. The multi-order model utilizes real data to make predictions regarding the unemployed as well as the employed population for the Northern states of India (J&K, HP, Punjab, Haryana) with an average absolute error less than 21% and 3%, respectively. When compared to the actual data, the fractional order model better captures the characteristics of the unemployed population than the integer order model. The fractional-order model exhibits lower RMSE, MAE and MAPE values and higher correlation coefficient (r) value.

In this study, the problem of synchronization of heterogeneous-driven response networks is introduced by using variable structure control strategy. First, the mathematical model of the heterogeneous-driven response network is described, and the development history of variable structure control is outlined. Second, considering the coupling time delay of the heterogeneous-driven response network, two types of variable structure controls are designed based on sliding mode control and soft variable structure control theories. Finally, the effectiveness of the two variable structure controls is verified through numerical simulation experiments, and a discussion on their advantages and disadvantages is provided.

Inspired by the phenomenon that individuals tend to make decisions after gaining enough experience from past interactions, heterogeneous influence adjustment considering historical strategy mechanism is proposed. This mechanism adjusts the node’s strategy propagation influence by comparing the multi-round game income with that of selected neighbors. The results of the spatial prisoner’s dilemma game indicate that cooperation is substantially enhanced when the mechanism strength factor $\alpha$ and the memory length m are suitably configured. Specifically, the mechanism strength factor $\alpha$ exhibits a positive correlation with the frequency of cooperative behavior because stronger incentives or deterrents more effectively influence individuals to adopt cooperative strategies. Furthermore, the extended memory length m contributes to an enhancement in the network’s cooperation fraction. Longer memory length diminishes the fluctuation magnitude of the cooperation fraction because longer memory allows individuals to make more informed decisions based on a more comprehensive understanding of past interactions, leading to more stable and sustained cooperative behavior.

Considering that rumors propagation is affected by many factors in real life, based on the SIRS infectious disease model in complex networks, an extended ISRI rumor propagation model is proposed by using the probability function to define the influence mechanisms such as trust mechanism, and suspicion mechanism. First, dynamic equations are established for homogeneous and heterogeneous networks, and the rumor and rumor-free equilibrium points in the two networks are analyzed, respectively. Then, the basic reproduction number $R_0$ is obtained by using the next generation matrix and derivative calculation methods. Next, the lyapunov function is constructed to discuss the local stability and global stability of the equilibrium point, and the influence of different parameters on the basic reproduction number $R_0$. In addition, we selected ER network and BA network and found that population flow has a significant impact on the speed and scale of rumor propagation. At the same time, the trust mechanism can improve the propagation speed and scale, while the skepticism mechanism can inhibit the propagation speed, and it is more obvious in the BA network. The interaction between these mechanisms further affects the propagation characteristics of rumors in the network.