# Information Propagation and Bionic Evolution Control of the _Sebar_ Model in a Swarm System
## 1 Introduction  
In nature, complex systems are ubiquitous. Biological groups such as bird flocks, fish schools, and ant colonies are composed of numerous individual entities or subsystems [1, 2]. Influenced by individual interactions or local regulation, these groups exhibit highly coordinated and orderly complex behaviors at the macroscopic level, sometimes even demonstrating a higher level of \"intelligence\" [3, 4]. As an important field of interdisciplinary research, complex systems theory and the study of swarm behaviors have garnered sustained attention from scholars in sociology, biology, computer science, and cybernetics. Extensive research has been conducted on core issues such as the stimulation, convergence, and emergence of swarm behaviors. However, previous studies paid less attention to the formation mechanisms of swarm consensus behaviors. And, information propagation and complex networks provide an effective means of representing the transition from simple to complex behaviors in these systems [5]. In addition, the network's structural properties have always been a focal point of network science [6, 7]. In recent years, with the continuous advancement of network science research, the study of the effects of propagation dynamics on networks has garnered sustained attention [8, 9]. This includes the spread of viruses among human populations [10], the transmission of computer viruses over networks [11], the dissemination of advertisements on the internet, and the spread of rumors on social networks [12]. Therefore, researchthe mechanisms and principles of information propagation is crucial, leading to the development of various propagation models, such as the SETQR model [13], which reflects rumor spreading and public opinion. Moreover, it is practically significant to implement effective measures to intervene in or regulate the information propagation process to guide it in the desired direction [14]. Examples include controlling high-impact nodes, managing critical edges, spreading counter-rumor information as part of rumor control strategies, and non-smooth control on a reaction diffusion networks [15]. From another perspective, information propagation and regulation on networks can be viewed as a complex group behavior involving both conflict and cooperation. As a challenging issue, it is important to study how \"cooperation\" emerges and is maintained within groups [16]. Evolutionary game theory on complex networks provides a crucial framework for investigating the evolution of cooperation and strategy competition [17; 18]. The main contributions of this paper involve the following aspects. From the information dissemination of the swarm system, a model of information transitions based on individual states is proposed to explain the consistency of swarm behaviors. In addition, a stability analysis of information propagation is also given. Different to previous studies, a repeated prisoner's dilemma game based on controllable individuals is proposed and the regulation of information transition and state evolution is realized. The simulation results demonstrate the influence of parameters and the validity of the proposed methods. The structure of this paper is organized as follows: In Section 2, the individual state transition rules and the \(SEBAR\) model are designed using mean-field theory. In Section 3, the equilibrium points are calculated and a basic reproduction of the amount of information propagation is provided. In Section 4, the stability of the equilibrium points is proved using the Jacobian matrix and the Routh-Hurwitz criterion. In Section 5, to achieve regulation of the information dissemination process, a repeated prisoner's dilemma game based on controllable individuals within the framework of the soft control of bionic swarm behavior is proposed, which integrates evolutionary game behavior into the complex networks of the swarm system.