With the continuous development of modern technology, online social networks (OSNs) have gradually become essential mediums for people to exchange opinions, which contributes to large-scale information propagation in different forms including rumor, awareness, influence and so on. It is well-known that the ubiquitous information propagation will significantly affect human daily life in turn. In particular, the spread of positive influence involving a certain city will help it absorb more talents and accelerate its productivity [1–3]. The awareness propagation about COVID-19 protection may save many human lives and reduce unnecessary economic losses [4–7]. And the rumor spreading about side effects of vaccine therapy can decrease the vaccination rates and hinder the effectiveness of disease interventions [8–10]. Therefore, exploring the dynamical process of information propagation is of great practical significance and has attracted much attention from the academic community. A large number of related work has followed on the mathematical modeling of information propagation dynamics, employing classical SIS, SIR and other derivative models. For example, Yu et al. [11] established an ignorance-discussant-spreader-remover rumor spreading model, and analyzed the local asymptotic stability of four equilibria. Yin et al. [12] took into account the emotion categories and choices of user communities on OSNs, and constructed an emotion-based susceptible-forwarding-immune propagation dynamic model to investigate the evolution process of emotional information and guide public sentiment. Zhu et al. [13] proposed a novel susceptible-latent-breaking-recovered propagation model to explore the dynamic behavior of industrial viruses in the SCADA system.

However, a considerable amount of previous work assumes that only single information propagates on OSNs, which is far beyond the practical case. In view of the individuals’ differences including nationality, ethnicity, education and personality, their opinions towards a specific event is generally diverged into two opposite sides, which can be considered as positive information and negative information. Typically, different groups of people always have different evaluations on a city, where local residents tend to prefer good evaluations, while nonresidents may make bad evaluations due to their terrible experiences. Then good evaluations promote the spread of positive influences, and bad evaluations lead to the spread of negative influences [14–16]. The effectiveness of COVID-19 prevention measures, such as vaccination and physical distancing, also simultaneously spreads through the crowd in dual forms of rumor and truth [17–21]. In particular, the government will disclose the truth about the effectiveness of prevention measures, and the supporters may actively promote the spread of truth. However, some opponents inevitably worry about the side effects of prevention measures and spread rumors. More attention should be paid to the spreading process of competitive information, so that the interaction between multiple information can more truly reflect the complexity of information propagation on OSNs. On the other hand, OSNs also form a multi-layer structure due to the clustering nature of the population, and the coupling between any layers will significantly affect the dynamic trend and steady-state prevalence of information propagation compared to single-layer networks. Antonopoulos et al. [22] studied the opinion dynamics over multiplex networks with heterogeneous confidence thresholds and general initial opinion distributions. Through probability analysis, they derived the analytical expressions for the critical thresholds of confidence bound considering certain regularity conditions of the networks, which reveals the significant impacts of structural multiplexity and initial distribution on the critical thresholds. Furthermore, their numerical simulations also explored the consensus behavior of agents in diverse network topologies. Yin et al. [23] constructed an environment-based susceptible-forwarding-immune model of information propagation on multi-platform networks, which reflects the complexity of actual social network topologies and the interaction between various social platforms. They also parameterized the model using COVID-19 data, and conducted the parameter sensitivity analysis. Wang et al. [24] investigated the multi-lingual SIR dynamic process of rumor spreading, which regarded each language community as a layer, and establish the rumor cross-transmitted mechanism in different lingual environments. Both the basic reproduction number and the stability of the rumor-free/endemic equilibrium were systematically discussed. These studies reveal many new characteristics how multi-layer network topologies intervene information propagation trend. However, although lots of work has separately investigated the potential effects of multiple information coexistence and multi-layer network topology, the attempts to synthesize the two aspects may reveal more interesting conclusions, which have not been well explored. Therefore, in order to more comprehensively explore the information propagation dynamics on OSNs, an improved model taking both the multiple information interaction and multi-layer interconnected networks needs to be further developed and analyzed.

Actually, individuals in OSNs are not only the carriers of information propagation, but also take certain adaptive behaviors according to the information prevalence. The herd behavior that individuals tend to follow others when making decisions is one of the most common social behaviors, and has recently attracted substantial attention from the research community [25–28]. However, to the best of our knowledge, there is few work to introduce individual herd behavior into the multiple information propagation on multi-layer networks, where herd behavior may change the mechanism of state transition and thus determine the dominance relationship between multiple information. Under the framework of multiple information propagation, each individual may consider both the local situation of its neighbors and the global situation of all individuals before deciding which type of information to spread. This adaptive herd behavior will affect the coupling characteristics between multiple information, and interfere with the evolution trend and prevalence of various information. Therefore, the consideration of individual herd decision-making behavior will also improve our understanding of information propagation dynamics.

One of the primary purposes for modeling information propagation is to control the spreading processes of interest. There are two main streams of control strategies, namely, heuristic control and optimal control [29–33]. Heuristic control strategy employs the real-time information prevalence to construct adaptive control function, aiming to effectively suppress/promote the specific information spreading, while optimal control strategy constructs the adaptive control function to maximize/minimize a well-designed objective function. Lin et al. [34] established a novel SWIR model with uncertain mental state to describe the spreading process of fraudulent information, and further formulated an optimal problem of two synergistic control strategies to minimize the total cost constraint involving individual losses and control resource consumption. Li et al. [35] proposed a multifactor-based information model considering information content and node characteristics, and also constructed an efficient heuristic algorithm of seed nodes selection to solve the targeted influence maximization problem. Wang et al. [36] proposed a novel computational model to present the temporal dynamics of the positive and negative information spread, and devised a nonlinear feedback control mechanism to perform three synergetic intervention strategies with minimal system expenses. During multiple information propagation, our control strategies are aimed at promoting the diffusion of positive information and suppressing the outbreak of negative information. Due to the resource limitations in reality, decision-makers always attempt to achieve the most powerful suppress performance of negative information with the lowest control intensity, which forms a trade-off between control cost and propagation cost. This trade-off can be formulated as an optimization problem, and the corresponding optimal solutions need to be further discussed, which can provide theoretical guidance and reference significance for practical resource scheduling.

Motivated by the above analysis, we aim to develop a novel framework that can integrate competitive information propagation, multi-layer interconnected networks and individual adaptive behavior. Our main contributions are summarized as follows.

• We propose a novel competitive information propagation model on multi-layer interconnected networks. This model can not only efficiently describe the competing phenomenon of positive and negative information coexistence in the real world, but also analyze the correlation between information propagation and multi-layer topology in OSNs;

The rest of this paper is organized as follows. In Section 2, a competitive information propagation model is formulated within the framework of multi-layer interconnected network. Stability analysis of information-free and information-endemic equilibria is carried out in Section 3. The formulation of optimal control problem is conducted to discuss the existence, analytic formulation and uniqueness of the optimal solution in Section 4. Extensive numerical simulations are conducted to verify our presented theoretical results in Section 5. Some conclusions are drawn in Section 6.

In this section, we formulate a spreading model SI ^A I ^B HR of competitive negative and positive information on multi-layer interconnected networks.

For simplicity, we here consider a two-layer interconnected network and denote them as layers 1 and 2 respectively, whose relevant conclusions can be easily generalized to the other cases of multi-layer networks with more layers. These two layers can be depicted as different countries, cities or communities in the real world. The individuals on two respective layers belong to different groups. Each individual has both the neighbors within the same layer and the acquaintances within other layer, which contributes to the intra-layer and inter-layer edges. Then denote the intra-layer average degree of the layer 1 and 2 as k 1 and k 2 respectively. And the inter-layer average degree from layer 1 to layer 2 is also denoted as k 12 , and the similar definition holds for k 21 . The diagram of multi-layer interconnected networks is shown in Figure 1A.

During the spreading dynamics SI ^A I ^B HR, all individuals are divided into five status compartments: Susceptible (S), Infected by positive information (I ^A ), Infected by negative information (I ^B ), Hesitate (H) and Recovered (R). Specifically, S state represents an individual who is unknown to any type of information but potentially infected via its intra-layer or inter-layer edges; I ^A state represents a spreader who is infected with positive information; I ^B state represents a spreader who is infected with negative information; H state represents an individual who has received two types of information but is hesitant to become which one; R state represents an individual who has no interest in spreading either positive or negative information. We further define the state density of S, I ^A , I ^B , H, R in layer 1 and 2 at time t as S ₁(t), S ₂(t), I 1 A ( t ) , I 2 A ( t ) , I 2 B ( t ) , I 2 B ( t ) , H ₁(t), H ₂(t), R ₁(t), R ₂(t), respectively.

Considering that the dynamic process of information propagation has the same time scale as the evolution of involved population numbers, we introduce the birth and mortality rates of population dynamics. And the birth rate and mortality rate of our dynamics are presented as ξ and δ respectively. Denote R + 5 the nonnegative cone and its lower dimensional face. Thus, we can study the SI ^A I ^B HR model in the following feasible region Ω = S i t , I i A t , I i B t , H i t , R i t ∈ R + 5 S i t + I i A t + I i B t + H i t + R i t ≤ ξ δ . (2.1)

The birth rate will increase the quantitative basis of susceptible individuals, which can potentially raise the infection risk in the population and promote the information prevalence. And the opposite condition holds for the mortality rate. The relative size of birth and mortality rates determines the evolutionary trend of population numbers. In real life, both birth and mortality rates have two meanings: 1) For the birth rate, on the one hand, it refers to natural birth rate, which means that the initial population can give birth to newborns and these newborns can participate in the information propagation; on the other hand, it also refers to immigrated birth rate, which means that the external population may move to the regions where the initial population is located due to work, education and tourism. After their immigration, they can also participate in the information propagation. 2) For mortality rate, on the one hand, it refers to natural mortality rate, which means that some individuals in the initial population will die due to accidents and old age, and these deaths may be excluded for the information propagation; on the other hand, it also refers to emigration mortality rate, which means that some initial individuals may move to other regions for diverse reasons, and their emigration makes them no longer involved in the information propagation. Taking the propagation process in layer 1 as an example, the state transition mechanism of SI ^A I ^B HR model can be described as.

• Some newborn individuals will be filled into the network with birth rate ξ to participate in the information propagation process. Without loss of generality, these newborn individuals are assumed to be susceptible; Individuals in all five compartments are removed from the network with mortality rate δ;

The dynamic evolution of information propagation for layers 1 and 2 can be derived as d S 1 t d t = ξ − k 1 λ 1 S 1 t I 1 A t − k 1 λ 2 S 1 t I 1 B t − k 12 κ λ 1 S 1 t I 2 A t − k 12 κ λ 2 S 1 t I 2 B t − δ S 1 t d I 1 A t d t = k 1 λ 1 S 1 t I 1 A t + k 12 κ λ 1 S 1 t I 2 A t + ε 1 A t H 1 t − k 1 θ 1 I 1 A t I 1 B t − k 12 κ θ 1 I 1 A t I 2 B t − δ + μ I 1 A t d I 1 B t d t = k 1 λ 2 S 1 t I 1 B t + k 12 κ λ 2 S 1 t I 2 B t + ε 1 B t H 1 t − k 1 θ 2 I 1 B t I 1 A t − k 12 κ θ 2 I 1 B t I 2 A t − δ + μ I 1 B t d H 1 t d t = k 1 θ 1 I 1 A t I 1 B t + k 12 κ θ 1 I 1 A t I 2 B t + k 1 θ 2 I 1 B t I 1 A t + k 12 κ θ 2 I 1 B t I 2 A t − ε 1 A t + ε 1 B t + δ + μ H 1 t (2.2) d S 2 t d t = ξ − k 2 λ 1 S 2 t I 2 A t − k 2 λ 2 S 2 t I 2 B t − k 21 κ λ 1 S 2 t I 1 A t − k 21 κ λ 2 S 2 t I 1 B t − δ S 2 t d I 2 A t d t = k 2 λ 1 S 2 t I 2 A t + k 21 κ λ 1 S 2 t I 1 A t + ε 2 A t H 2 t − k 2 θ 1 I 2 A t I 2 B t − k 21 κ θ 1 I 2 A t I 1 B t − δ + μ I 2 A t d I 2 B t d t = k 2 λ 2 S 2 t I 2 B t + k 21 κ λ 2 S 2 t I 1 B t + ε 2 B t H 2 t − k 2 θ 2 I 2 B t I 2 A t − k 21 κ θ 2 I 2 B t I 1 A t − δ + μ I 2 B t d H 2 t d t = k 2 θ 1 I 2 A t I 2 B t + k 21 κ θ 1 I 2 A t I 1 B t + k 2 θ 2 I 2 B t I 2 A t + k 21 κ θ 2 I 2 B t I 1 A t − ε 2 A t + ε 2 B t + δ + μ H 2 t (2.3) where ε 1 A t = β 1 ω 1 I 1 A t + 1 − ω 1 1 2 I 1 A t + I 2 A t , ε 1 B t = β 1 ω 1 I 1 B t + 1 − ω 1 1 2 I 1 B t + I 2 B t , (2.4) ε 2 A t = β 2 ω 2 I 2 A t + 1 − ω 2 1 2 I 1 A t + I 2 A t , ε 2 B t = β 2 ω 2 I 2 B t + 1 − ω 2 1 2 I 1 B t + I 2 B t . (2.5)

For Eqs 2.4, 2.5, the local information prevalence is equivalent to the density of information spreaders in the layer where individuals reside, while the global information prevalence is defined as the average density of information spreaders in two layers. ω ₁ and ω ₂ are the weighted coefficients of local prevalence of negative/positive information in layer 1 and 2. β ₁, β ₂ are the scaling factors. Without loss of generality, we set β ₁ = β ₂ = 1 here.

In this section, we firstly study the basic reproduction number of dynamical systems (2.2) and (2.3) through the next-generation matrix method [37]. Secondly, the existence and stability of information-free equilibrium and information-endemic equilibrium are discussed by means of analysis technique, Routh-Hurwitz judgment, Lyapunov method and LaSalle’s invariance principle.

In order to facilitate the calculation of the basic reproduction number, denote X t = I 1 A t , I 2 A t , I 1 B t , I 2 B t , H 1 t , H 2 t , S 1 t , S 2 t T . (3.1)

Then we can rewrite the dynamical systems (2.2) and (2.3) as d X t d t = F X t − V X t (3.2) where F = k 1 λ 1 S 1 t I 1 A t + k 12 κ λ 1 S 1 t I 2 A t k 2 λ 1 S 2 t I 2 A t + k 21 κ λ 1 S 2 t I 1 A t k 1 λ 2 S 1 t I 1 B t + k 12 κ λ 2 S 1 t I 2 B t k 2 λ 2 S 2 t I 2 B t + k 21 κ λ 2 S 2 t I 1 B t 0 0 0 0 , (3.3) V = k 1 θ 1 I 1 A t I 1 B t + k 12 κ θ 1 I 1 A t I 2 B t + δ + μ I 1 A t − ε 1 A t H 1 t k 2 θ 1 I 2 A t I 2 B t + k 21 κ θ 1 I 2 A t I 1 B t + δ + μ I 2 A t − ε 2 A t H 2 t k 1 θ 2 I 1 B t I 1 A t + k 12 κ θ 2 I 1 B t I 2 A t + δ + μ I 1 B t − ε 1 B t H 1 t k 2 θ 2 I 2 B t I 2 A t + k 21 κ θ 2 I 2 B t I 1 A t + δ + μ I 2 B t − ε 2 B t H 2 t ε 1 A t + ε 1 B t + δ + μ H 1 t − θ 1 + θ 2 k 1 I 1 A t I 1 B t − κ k 12 θ 1 I 1 A t I 2 B t + θ 2 I 1 B t I 2 A t ε 2 A t + ε 2 B t + δ + μ H 2 t − θ 1 + θ 2 k 2 I 2 A t I 2 B t − κ k 21 θ 1 I 2 A t I 1 B t + θ 2 I 2 B t I 1 A t k 1 S 1 t λ 1 I 1 A t + λ 2 I 1 B t + κ k 12 S 1 t λ 1 I 2 A t + λ 2 I 2 B t + δ S 1 t − ξ k 2 S 2 t λ 1 I 2 A t + λ 2 I 2 B t + κ k 21 S 2 t λ 1 I 1 A t + λ 2 I 1 B t + δ S 2 t − ξ (3.4)

Obviously, there exists an information-free equilibrium E 0 = X ( ∞ ) = 0,0,0,0,0,0 , ξ δ , ξ δ . Furthermore, the Jacobian matrices of F ( X ) and V ( X ) at E ₀ are D F E 0 = F 0 0 0 , D V E 0 = V 0 J 1 J 2 , (3.5) where F = k 1 λ 1 ξ δ k 12 κ λ 1 ξ δ 0 0 k 21 κ λ 1 ξ δ k 2 λ 1 ξ δ 0 0 0 0 k 1 λ 2 ξ δ k 12 κ λ 2 ξ δ 0 0 k 21 κ λ 2 ξ δ k 2 λ 2 ξ δ , V = δ + μ 0 0 0 0 δ + μ 0 0 0 0 δ + μ 0 0 0 0 δ + μ . (3.6)

Therefore, the next-generation matrix can be calculated as Γ = F V − 1 = k 1 λ 1 ξ δ δ + μ k 12 κ λ 1 ξ δ δ + μ 0 0 k 21 κ λ 1 ξ δ δ + μ k 2 λ 1 ξ δ δ + μ 0 0 0 0 k 1 λ 2 ξ δ δ + μ k 12 κ λ 2 ξ δ δ + μ 0 0 k 21 κ λ 2 ξ δ δ + μ k 2 λ 2 ξ δ δ + μ . (3.7)

The basic reproduction number R 0 is defined as the spectral radius of matrix Γ,namely, R 0 = ρ Γ = ρ F V − 1 = ξ max λ 1 , λ 2 2 δ δ + μ k 1 + k 2 + k 1 − k 2 2 + 4 κ 2 k 12 k 21 . (3.8)

Combined the above analysis, the stability properties of the information-free equilibrium E ₀ and information-endemic equilibrium E* are given in the following theorems.