Hyper-S2IR: a model for characterizing higher-order interactions and dynamics in public opinion dissemination

Abstract

Traditional public opinion diffusion models generally assume interactions between individuals as binary pair-wise effects, which struggle to capture the higher-order complexities of multi-group interactions in social networks—such as group discussions in WeChat and topic reposting on Weibo. Moreover, these models fail to adequately depict the nonlinear trust accumulation mechanisms and individual heterogeneity inherent in the diffusion process. Therefore, the paper proposes a hypergraph-based Hyper-S2IR model for disseminating public opinion. The “Goebbels effect” is operationalized by leveraging the hypergraph structure: a susceptible node’s risk of infection is proportional to its hyperdegree, mathematically representing the cumulative exposure to information from multiple sources within different hyperedges. Our model introduces two types of communicators (HI₁ and HI₂) with different motivations and capabilities, thereby systematically depicting the inherent heterogeneity of the communication group. Through theoretical derivation, we derive a novel basic reproduction number R₀ that explicitly incorporates the hyperdegree distribution of the hypergraph. This R₀ provides a threshold for dissemination dynamics: When R₀ > 1, the public opinion will continue to spread and converge to a stable public opinion prevalence equilibrium point; when R₀ < 1, the public opinion will gradually disappear. Critically, the expression for R₀ reveals how higher-order group interactions, encoded in the hyperdegree, fundamentally alter the spreading threshold compared to traditional pairwise networks. Numerical simulations verify the theoretical conclusions and demonstrate that the hypergraph structure significantly accelerates the spread and expands the scale of public opinion compared to traditional network structures. This work provides theoretical support and a quantitative basis for analyzing public opinion dissemination mechanisms and formulating intervention strategies.

1 Introduction

With the increasing penetration and rapid development of the Internet, social networking platforms such as Weibo and WeChat have become primary channels for public communication. Information spreads on these platforms at unprecedented speed, and both the diversity and volume of online content have expanded by orders of magnitude compared with earlier media environments. In this context, public opinion often emerges from the combined effects of social environments, individual psychology, and information transmission. Accordingly, understanding, monitoring, guiding, and responding to public opinion has become a major challenge for individuals, institutions, and governments. Research on public opinion dissemination, therefore, carries substantial theoretical significance and practical value: it not only enhances the ability to interpret communication phenomena in the new-media era but also provides scientific decision support and actionable guidance for social governance, organizational management, and national development [1].

At its core, public opinion dissemination is a process of information transmission, whose propagation mechanism shares notable similarities with infectious disease spread. Consequently, early studies of public opinion diffusion often drew on classical epidemic models. The compartmental modeling framework was first proposed by Kermack and McKendrick [2], who studied outbreaks such as the plague and assumed homogeneous mixing, instantaneous contact, and no dependence on interaction history. Building on this framework, a series of classic epidemic models have been developed, including SI, SIS, SIR, SIRS, and SEIR models, among others [3–5].

A seminal model for public opinion dissemination is the Daley–Kendall model proposed in the 1960s [6], whose basic mechanism parallels the SIR model; Maki and Thomson further refined it in 1973 [7]. With growing recognition that network structure shapes diffusion dynamics, researchers subsequently proposed public opinion dissemination models based on complex networks. In 2001, Pastor-Satorras and Vespignani analyzed spreading processes on regular, random, and small-world networks and provided theoretical evidence for the existence of non-zero thresholds [8]. Meanwhile, Zanette [9, 10] was among the first to investigate public opinion spreading on complex networks, arguing that a threshold θ exists in small-world networks: when spreading parameters fall below θ, public opinion will eventually die out. These studies collectively underscore that network topology can significantly influence dissemination dynamics.

Despite these advances, most existing models still rely on a pairwise-interaction paradigm, assuming that individual interactions occur only between two nodes. In real online environments, however, interactions frequently take place at arbitrary scales, ranging from one-to-one communication to information diffusion within groups of varying sizes—for example, discussions in WeChat group chats or the collective reposting and commenting around Weibo topics. In addition, public opinion diffusion is often accompanied by the so-called “Goebbels effect,” whereby individuals may initially doubt a message but gradually increase their trust after repeated exposure from different sources, eventually accepting it as true [11, 12]. Repeated exposure and group interactions are thus key drivers of state transitions in public opinion dynamics. Nevertheless, prevailing models exhibit clear limitations in representing these mechanisms. First, by remaining grounded in pairwise interactions, they struggle to describe higher-order and synchronous diffusion within groups. Second, they often assume fixed transition probabilities, thereby oversimplifying the nonlinear process through which trust is dynamically reinforced via repeated exposure, as conceptualized by the Goebbels effect.

The Goebbels effect reveals a fundamentally nonlinear process of trust formation at both individual and group levels. Cognitively, it can be interpreted as a dynamic evidence-accumulation mechanism, in which repeated exposure nonlinearly shifts beliefs by reducing cognitive inertia and status quo bias in decision-making [13]. From a modeling perspective, it also suggests a “major–minor factor” structure in trust formation: exposure frequency serves as the primary driver, while individual sensitivity modulates its impact [14]. From a social-dynamics viewpoint, the effect aligns with the logic that moral alignment fosters group consensus, emphasizing how repeated micro-level interactions can lead to macro-level consensus formation [15]. Therefore, modeling the Goebbels effect is not merely a refinement beyond simple probabilistic transitions, but a more faithful representation of the cognitive and social mechanisms underlying public opinion formation [16].

Furthermore, many models treat spreaders as homogeneous, overlooking heterogeneity in motivation, capability, and social influence. Consequently, building an integrated framework that jointly captures group interactions, nonlinear trust accumulation, and spreader heterogeneity has become a central challenge in modeling public opinion diffusion in real-world networks [17]. The homogeneity assumption is an oversimplification that weakens model realism and predictive validity: in practice—especially for contentious topics—spreaders differ markedly in emotional tone, motivation, and persuasive power. Emotionally aroused spreaders may repost rapidly and reflexively, whereas deliberative spreaders are more likely to fact-check and engage in reasoned discussion. Such heterogeneity fundamentally alters macroscopic diffusion outcomes, affecting diffusion speed, scale, and persistence, and can yield phenomena such as multi-peaked diffusion curves or the sustained coexistence of competing narratives—patterns that homogeneous-mixing models cannot capture. Hence, moving beyond homogeneity to explicitly model communicator heterogeneity is essential for developing more realistic and predictive diffusion models.

Hypergraphs [18], as an important extension of graph theory for modeling higher-order interactions, allow a single hyperedge to connect multiple nodes. This property makes hypergraphs well-suited for representing group interaction units such as WeChat groups and topic communities, thereby enabling a more accurate description of synchronous information diffusion within groups [19]. Motivated by this, and to better reflect intrinsic differences among communicators, this paper draws on the S2IR model [20] and explicitly distinguishes a second type of disseminator (I₂). A new hypergraph-based public opinion dissemination model, termed Hyper-S2IR, is proposed, and its dynamical behavior is examined through theoretical analysis and simulation-based validation.

2 Related work

2.1 Hypergraph and hypergraph applications

In hypergraphs, a hyperedge can simultaneously connect an arbitrary number of nodes, enabling a natural representation of multi-member group structures such as WeChat groups, online communities, and collaborative teams. Owing to this expressive power, hypergraphs have been widely applied in social network analysis, recommender systems, and knowledge graphs [21]. In dissemination dynamics, hypergraphs provide a principled framework for describing real-world group diffusion, where information often spreads synchronously within groups rather than solely through pairwise contacts. In recent years, classical spreading models have been successfully extended to hypergraph settings [22–24]. However, although these studies highlight the advantages of hypergraphs for modeling population-level transmission, most of them retain the classical SIS/SIR-style compartmental architecture and thus largely overlook intrinsic heterogeneity among transmitters.

Recent progress in graph learning further supports the value of hypergraph-based modeling. For tasks that require complex relational reasoning, such as fact verification, approaches that strengthen structural connectivity have been shown to yield superior performance [25]. Similarly, models that explicitly capture higher-order dependencies often achieve improved accuracy and robustness, particularly in low-label regimes [26]. These findings jointly suggest that incorporating higher-order relational structures is not only theoretically well motivated but also empirically beneficial for modeling and inference in complex information environments.

Figure 1 illustrates an example of a hypergraph and hypergraph dissemination.

FIGURE 1

In Figure 1A, the node set V = {v₁, v₂, … , v₉} and the hyperedge set E = {E₁, E₂, E₃, E₄}. E₁ = {v₁, v₂, v₉}, E₂ = {v₂, v₃, v₄, v₅, v₆}, E₃ = {v₆, v₇}, and E₄ = {v₇, v₈, v₉}. Consequently, the hyperdegree of v₁, v₃, v₄, v₅, and v₈ is 1, while that of v₂, v₆, v₇ and v₉ is 2. Figure 1B illustrates an example of hypergraph dissemination, where red nodes represent infected nodes and blue nodes denote susceptible nodes. The probability that the infected node v₆ infects the susceptible node v₇ through a hyperedge of size 2 is β₂. If a hyperedge E₄ of size 3 contains two infected nodes v₈ and v₉, then they will infect the unique susceptible node through this hyperedge, with a probability of β₃.

2.2 Fundamental theory of dissemination models

With continued advances in understanding information dissemination mechanisms, communication models have evolved from early homogeneous and deterministic frameworks toward more sophisticated systems that can represent the heterogeneity and dynamism of complex social environments. Recent studies have made substantial progress in developing integrated analytical frameworks that incorporate network heterogeneity and behavioral complexity. Nevertheless, several issues remain open. First, although complex-network and multilayer-network approaches can effectively characterize binary relationships between nodes [27, 28], most models are still fundamentally grounded in pairwise interactions and thus have difficulty capturing the synchronous group interactions that frequently occur in real-world settings. Second, while psycho-behavioral coupling mechanisms have been introduced to partially account for individual differences, many traditional models still treat communicators as a largely homogeneous population, providing insufficient characterization of within-group differentiation in motivation, emotional tendencies, and behavioral patterns [29, 30].

In real-world public opinion dissemination—especially during sudden public incidents, social issue debates, or political elections—the information environment is rarely singular. Instead, it is typically composed of multiple competing, opposing, or coexisting viewpoints, narratives, and emotional inclinations. The S2IR model proposed by Wang et al. [31] offers a useful reference for representing such heterogeneity by distinguishing two categories of disseminators.

The state transition diagram for the S2IR model is shown in Figure 2.

FIGURE 2

The differential equation form of the S2IR model is shown in Equation 1.

In the equation, S(t), I_A(t), I_B(t), and R(t) denote the densities of the four population categories at time t, satisfying S(t)+ I_A(t)+ I_B(t)+ R(t) = 1. β represents the transmission rate, γ denotes the immunity rate, while σ_AB and σ_BA represent the conversion rates between transmission and immunity states.

Mathematical formalization of the Goebbels effect. To mechanistically capture the “Goebbels effect”—where repeated exposure induces cumulative trust—a general formalization is introduced. The key premise is that the total infection pressure experienced by a susceptible individual i, denoted by Λ_i(t), is a monotonically increasing function of the effective exposure count N_i(t). The specific formula is shown in Equation 2.

Here, N_i(t) represents the effective and independent number of information exposures that individual i has experienced by time t. The function f (·) is a monotone increasing trust-accumulation function that maps exposure counts to a trust level, f (0) = 0. The term ρ(t) denotes the baseline infection pressure per exposure, reflecting factors such as message virality or contextual intensity.

To situate the present study within the literature and clarify its contributions, Table 1 summarizes representative opinion dissemination models along three dimensions: underlying network structure, treatment of spreader types, and primary analytical focus. As shown, prior work has made notable progress in modeling heterogeneous spreaders on graphs [17, 28, 31] and homogeneous spreaders on hypergraphs [22, 32]. However, a critical gap remains in frameworks that can simultaneously capture (i) higher-order group interactions naturally represented by hypergraphs and (ii) intrinsic heterogeneity among communicators. The Hyper-S2IR model is developed to address this gap by integrating these two essential aspects, thereby providing a more nuanced and realistic characterization of public opinion dissemination.

3 Hyper-S2IR model

3.1 Hyperedge-degree dissemination

In the hypergraph, the hyperdegree of node v_i represents the breadth of its participation in group interactions, defined as the set of all hyperedges to which the node belongs. The hyperdegree vector of node v_i is represented as an (m-1)-dimensional vector, as shown in Equation 3.

Among them, k_i^(m) represents the mth degree of the node. It represents the number of hyperedges of size m for the node, where m = 2, 3, …, M. The hyperdegree is arranged in ascending order of each degree. For example, [5, 2, 0, …, 0] is placed before [6, 2, 0, …, 0], [2, 3, 1, …, 0] is placed before [2, 3, 2, …, 0], and [5, 1, 2, 0, 1, …, 0] is placed after [5, 1, 0, 2, 1, …, 0] [32].

In Figure 1A, the excess degrees of v₁ and v₈ are [0,1,0,0], the excess degree of v₂ is [0,1,0,1], the excess degrees of v₃, v₄, and v₅ are [0,0,0,1], and the excess degrees of v₆, v₇, and v₉ are [1,0,0,1], [1,1,0,0], and [0,2,0,0] respectively. According to the above sorting method and arranging in ascending order of excess degree numbers, then K₁ = [0,0,0,1], In addition, K₂ = [0,1,0,0], K₃ = [0,1,0,1], K₄ = [0,2,0,0], K₅ = [1,0,0,1], and K₆ = [1,1,0,0].

3.2 Model assumptions

The meanings of the various parameters involved in the Hyper-S2IR dissemination model are shown in Table 2.

The state transition diagram of the Hyper-S2IR dissemination model is shown in Figure 3.

FIGURE 3

Hyper-S2IR model dynamic equations are as follows:

Here, HS_Ki(t), HI_1Ki(t), HI_2Ki(t), and HR_Ki(t) denote the densities of the four types of state nodes with hyperdegree K_i at time t, respectively. α, β₁, β₂, γ, and ω are non-negative constant parameters.

The system of dynamic Equation 4 describes the rate of change for each node state. The physical interpretation of each equation is as follows:

dHS_Ki(t)/dt: The population decreases as nodes are infected by contact with HI₁ or HI₂ nodes in hyperedges (-||K_i||(Θ₁+Θ₂)HS), and increases as recovered nodes lose immunity (+ωHR).

dHI_1Ki(t)/dt: The population grows with new infections of susceptible nodes by HI₁ nodes (+α||K_i||HSΘ₁). It decreases through spontaneous recovery (-γHI₁) and state transitions to HI₂ due to competition (-β₁HI₁Θ₂), but can gain nodes converted from HI₂(+β₂HI₂Θ₁).

dHI_2Ki(t)/dt: This equation is symmetric to the HI₁ equation, describing the analogous dynamics for the second type of infected nodes.

dHR(t)/dt: The population increases from susceptible nodes that do not become spreaders after contact (+(1-α)||K_i||(Θ₁+Θ₂)HS) and from the recovery of both infected types (+γ(HI₁+HI₂)). It decreases as nodes lose immunity and re-enter the susceptible compartment (-ωHR).

The infection probability function Θ_n(t) denotes the probability that any given hyperedge points to an infected node, as shown in Equation 5.

Here, denotes the intensity of public opinion dissemination by infected nodes of degree K_i. Let M be the size of the largest hyperedge in the hypergraph, as shown in Equation 6.

Here, α_m denotes the infection rate of hyperedges of size m, signifies the probability at the current time step that an infected neighbor node of degree K_i is connected to a given node. .

P(K_i|K_j) denotes the conditional probability that a node with degree K_i is connected to a node with degree K_j, as shown in Equation 7.

Here, is the average hyperdegree. Therefore,

Obviously, , then

Here, P(K_i) is the hyperdegree distribution.

4 Model equilibrium point analysis

4.1 No-public opinion equilibrium point and proof of existence

4.2 Derivation of the basic reproduction number R₀

This paper employs a linearization analysis method for the infection subsystem to derive R₀. Near the equilibrium point without public opinion, the dynamical equations governing the infection variables are linearized, and the corresponding Jacobian matrix M is constructed. By calculating the maximum eigenvalue of this matrix, the dominant growth rate of the system is obtained, thereby defining the basic reproduction number.

Consider the system dynamics equations at the equilibrium points where HS_Ki = 1, HI_1Ki = 0, HI_2Ki = 0, and HR_Ki = 0. Express all variables as equilibrium values plus perturbation terms (e.g., HI_1Ki = 0+δHI_1Ki), retaining only first-order small terms. Since the proportion of susceptible individuals remains close to 1 near the no-public opinion equilibrium point, and its perturbation term is initially far smaller than the infection variable, HS_Ki can be approximated as the constant 1. Similarly, the cross-terms within the system of dynamical equations constitute higher-order infinitesimals when the infection variable is extremely small and may be neglected in preliminary linearized analyses.

The approximate equation for the dynamics of the infected subsystem is shown in Equation 13.

Record all infection-related variables as a 2z-dimensional state vector as shown in Equation 15.

After linearization, the dynamic system Equation 13 is transformed as shown in Equation 16.

Here, M denotes the Jacobian matrix of the system, which is a 2z × 2z block matrix whose form is as shown in Equation 17.

Here, A and D represent the “self-consistent dissemination” and self-dissipation components of HI_1Ki and HI_2Ki, respectively, while B and C denote the linear coupling of cross-branches.

Block A: The self-consistent dissemination and loss of the HI_1Ki on its own branch are described by the matrix elements as shown in Equation 18.

Block B and Block C: The cross-coupling between HI_2Ki and HI_1Ki, as revealed by the system of differential Equation 14, indicates no direct interaction coupling between them. The matrix values are thus expressed as in Equation 19.

Assuming the eigenvector is x = [x₁, … ,x_z]^T and the eigenvalue is λ, we can obtain the result as shown in Equation 21.

Let , and simplifying, as shown in Equation 22.

Substituting x_i into the derivation process of the definition of HS yields, as shown in Equation 23.

According to the threshold theorem for public opinion dissemination, when R₀ = 1, the dissemination of public opinion reaches a critical point. At this juncture, the eigenvalues λ of the Jacobian matrix’s characteristic equation in the dissemination dynamics equation set are equal to zero. In summary, the definition formula for R₀ is presented as shown in Equation 25.

When R₀ > 1, the system possesses the capacity for sustained transmission; when R₀ ≤ 1, the system ultimately returns to a state of no infection.

4.3 The public opinion equilibrium point and proof of existence

The equilibrium point of public opinion refers to the state in which, following prolonged system evolution, all state variables stabilize, and a proportionate number of infected individuals persist. This is characterized by the conditions HI_1Ki > 0 and HI_2Ki > 0 holding for all group i, reflecting the capacity of public opinion to sustain itself and endure over the long term within the network.

To calculate the epidemic equilibrium point of public opinion, set the derivatives of all variables with respect to time in the system of dynamic Equation 4 to zero, yielding the system of algebraic equations as shown in Equation 26.

Simplifying the system of Equation 26, we obtain:

The expressions for HI_1Ki and HI_2Ki in the simultaneous Equation 27 can be calculated as shown in Equation 28.

Combining HS_ki^*+ HI_1ki^*+ HI_2ki^*+ HR_ki^* = 1, calculate the equilibrium points of the system of equations. As shown in Equation 29.

Where ϴ₁^* and ϴ₂^* satisfy the self-consistency relationship as shown in Equation 30:

4.4 Theoretical analysis of the persistent circulation of public opinion

Let the state variable X be defined as shown in Equation 31.

The dynamic Equation 4 is converted as shown in Equation 32.

5 Stability analysis

6 Numerical examples

To demonstrate the model’s validity, numerical simulations were designed based on the system (4). The hyper-degree distribution follows a generalized power-law form: , where b represents the minimum value of ||K_i||. Assuming the range of ||K_i|| in the network spans from 1 to 200, where , . The numerical simulation experiment was set with initial conditions: S (0) = 0.96, I₁ (0) = 0.02, I₂ (0) = 0.01, and R (0) = 0.01.

6.1 Stability numerical simulation of the no-public opinion equilibrium point

Let α = 0.37, β₁ = 0.3, β₂ = 0.2, γ = 0.09, ω = 0.375, = 0.065. By Equation 25, we obtain R₀ = 0.958 < 1. According to Theorem 5.1, the no-public opinion equilibrium point of the system (7) is locally asymptotically stable.

Figure 4 shows the numerical simulation evolution of population state densities over time t for the system (4) under the condition R₀ < 1. The multiple faint lines represent individual simulation trajectories, while the thick solid lines denote the average evolution curves for corresponding variables. HS(t) first declines before rebounding, ultimately stabilizing at a non-zero value. HI₁(t) and HI₂(t) successively exhibit single peaks, with the peak of HI₂(t) l lagging markedly behind that of HI₁(t). HR(t) increases monotonically to a saturated state. The system ultimately converges to a stable equilibrium where HI₁(t) = HI₂(t) = 0 while HS(t) and HR(t) > 0, indicating the establishment of stable herd immunity during transmission. The evolutionary trends of each state variable align with theoretical analysis. The average curves exhibit smoothness with fluctuations within reasonable ranges, validating the model’s stability and convergence under different parameter implementations.

FIGURE 4

Figure 5 shows the density distribution of five population categories within system (4) through a heatmap. Where HS(t) exhibits a pronounced gradient decay, reflecting its continuous depletion during the transmission process. HI₁(t) exhibits a concentrated distribution, indicating an earlier infection peak with significant spatial clustering. HI₂(t) exhibits relatively dispersed high-density zones, suggesting a more gradual transmission process. HR(t) exhibits that the recovery phase dominates initially before levelling off. The overall infected state HI(t) density comprehensively reflects the spatiotemporal heterogeneity resulting from the superposition of these two infection states.

FIGURE 5

Numerical simulations demonstrate that when the basic reproduction number R₀ < 1, the no-public opinion equilibrium point of the system Equation 4 exhibits global attractiveness. Initial transmission proportions were set at (0.01, 0.05, 0.1, 0.2), observing the system’s state evolution trajectory over time.

As evident from the subplots in Figure 6, under parameter conditions where R₀ < 1, the no-public opinion equilibrium point exhibits global asymptotic stability. Regardless of the initial state, the phase trajectories of the system demonstrate a clear convergence trend in both two-dimensional and three-dimensional projections, ultimately stabilizing at the no-public opinion equilibrium point. The numerical results are consistent with Theorem 4.2 and Theorem 5.3. That is, when the basic reproduction number R₀ is less than 1, regardless of the initial proportion of the population affected by the rumors, the system ultimately converges to the no-public opinion equilibrium point.

FIGURE 6

6.2 Stability numerical simulation of the public opinion equilibrium point

Let α = 0.37, β₁ = 0.3, β₂ = 0.2, γ = 0.01, ω = 0.001, and = 0.065. By Equation 24, we obtain R₀ = 8.618 > 1. According to Theorem 5.2, when parameters ϴ₁* and ϴ₂* satisfy certain conditions, the public opinion equilibrium point is locally asymptotically stable.

Figure 7 shows the numerical simulation evolution of population state densities over time t for the system (4) under the condition R₀ > 1. HS(t) exhibits a typical exponential decay trend. As t increases, the population state density of HS(t) continuously diminishes before stabilizing at a non-zero steady state. HI₁(t) and HI₂(t) sequentially form single peaks, with the peak of HI₂(t) lagging significantly behind that of HI₁(t). HR(t) exhibits a monotonically increasing S-shaped growth trend, with a rapid initial growth rate that gradually slows before ultimately reaching a saturated state. The system ultimately converges to a stable equilibrium where HS(t), HI₁(t), HI₂(t), and HR(t) > 0. This equilibrium state indicates the establishment of a stable herd immunity pattern following the conclusion of transmission. The evolutionary trends of each state variable align with theoretical analysis. The average curves exhibit smoothness with fluctuations within reasonable bounds, validating the model’s stability and convergence under various parameters.

FIGURE 7

Figure 8 shows the density distribution of five population categories within system Equation 4 through a heatmap. HS(t) exhibits a gradient from top-left to bottom-right, indicating that susceptible individuals are progressively depleted over time, leading to a gradual decrease in density. HI₁ (t) exhibits a pronounced lateral band of high density at the midpoint of the timeframe, signifying rapid transmission and peak prevalence of HI₁ across a substantial proportion of nodes, followed by subsequent attenuation. The high-density regions for HI₂(t) emerge slightly later than those for HI₁ and display a more dispersed distribution, reflecting the asynchrony and heterogeneity of transmission. HR(t) density exhibits a color gradient from bottom-left to top-right, indicating that the density of recovered individuals accumulates over time, becoming dominant in the later stages. The high-density region of the total infected state HI (t) persists for an extended period, enabling infections to persist within the system and form distinct transmission peaks.

FIGURE 8

Numerical simulations demonstrate that when the basic reproduction number R₀ > 1, the public opinion equilibrium point of the system Equation 4 exhibits global attractiveness. Initial transmission proportions were set at (0.01, 0.05, 0.1, 0.2), observing the system’s state evolution trajectory over time.

As shown in the subplots of Figure 9, under parameter conditions where R₀ > 1, the evolutionary trajectory of the system ultimately stabilizes at the same public opinion equilibrium point regardless of the initial infection proportion. This indicates that the epidemic will not die out at this stage but instead establishes a persistent endemic state within the population.

FIGURE 9

6.3 Dynamics of public opinion dissemination: Hypergraph versus graph

Under the condition that the basic reproduction number R₀ < 1, the dissemination dynamics of system (4) were compared in both hypergraph and ordinary graph network structures.

As shown in Figure 10, under the condition that R₀ < 1, regardless of the network structure, the system trajectory ultimately converges towards the no-public-opinion equilibrium point. This numerically confirms the global stability of this equilibrium point within the parameter range. However, transmission within hypergraph structures exhibits characteristics of higher peaks, larger scales, and slower decay. This arises because hyperedges simulate collective interactions, enabling infections to simultaneously affect multiple susceptible nodes within a single time step, thereby amplifying the dissemination effect. By contrast, the pairwise interactions in ordinary graphs constrain both the speed and scope of transmission.

FIGURE 10

Under the condition that the basic reproduction number R₀ > 1, the dissemination dynamics of system Equation 4 were compared in both hypergraph and ordinary graph network structures.

As shown in Figure 11, under conditions where R₀ > 1, regardless of the network structure, the system trajectory ultimately converges towards the public opinion equilibrium point of public opinion prevalence. Comparative analysis reveals that higher-order interactions within networks significantly accelerate and amplify the dissemination of public opinion. In hypergraph structures, infection peaks are higher, peak attainment occurs more rapidly, and the system’s convergence towards equilibrium is prolonged.

FIGURE 11

To quantify the role of higher-order interactions in opinion spreading, this paper compare the Hyper-S2IR model on two network substrates: a hypergraph, where a hyperedge represents simultaneous group interaction, and a graph, where only pairwise contacts are allowed. Under identical parameter settings and numerical integration protocols, this paper extracts the following outcome metrics from the infection trajectory HI(t): (i) the basic reproduction number R₀; (ii) the infection peak HI_peak; (iii) the peak time t_peak; (iv) the final infection level HI_final = HI(t_end); and (v) the maximum propagation velocity V_max = max_t. Let α = 0.37, γ = 0.09, β₁ = 0.3, β₂ = 0.2, ω = 0.001. The comparison results are shown in Table 3.

Under the same parameter setting, the classical graph and the hypergraph exhibit markedly different spreading dynamics. In terms of threshold strength, R₀ = 156.0 on the graph but R₀ = 5.449 on the hypergraph, indicating that higher-order interactions rescale the effective spreading threshold. At the macroscopic level, the graph case reaches a later and higher peak and ends with a larger final infected level, whereas the hypergraph reaches its peak much earlier with HI_peak = 0.0962, HI_final = 0.0559, and a substantially different maximum propagation velocity. Overall, the hypergraph structure does not merely shift a single outcome but reshapes the threshold scale, peak timing, and growth-rate characteristics of propagation, leading to a distinct spreading profile compared with the classical graph.

6.4 Analysis of the dissemination impact of state change parameters

FIGURE 12

FIGURE 13

FIGURE 14

FIGURE 15

FIGURE 16

A global sensitivity analysis was conducted for {α, γ, β₁, β₂, ω} using a two-step pipeline. Morris screening was applied first, followed by Sobol variance-based decomposition using the total-effect index S_T.

Tables 4, 5 show that γ dominates the variance of key outputs, particularly peak timing, while is consistently the second most influential parameter. contributing notably to -related peak and final outcomes and remaining non-negligible for dynamics. Morris screening provides a consistent ranking, confirming that type-specific peak outputs better reveal the effects of the cross-type conversion and competition parameters.

7 Conclusion

7.1 Conclusions and discussion

Based on the construction and analysis of the Hyper-S2IR model rooted in hypergraph theory, this study provides a theoretical framework for explaining opinion diffusion under group interactions. The main contribution is the use of hypergraphs to represent higher-order social contacts, together with two categories of propagators to capture heterogeneity in motivation and influence, thereby modeling phenomena such as the “Goebbels effect” and incentive-driven competition in propagation. The basic reproduction number is derived analytically, yielding a clear survival threshold: when the threshold condition is not met, the system converges to the no-opinion equilibrium, whereas exceeding it leads to persistent diffusion. Comparative experiments between hypergraph and classical graph structures further show that higher-order interactions do not simply change one outcome, but rescale the effective threshold and reshape the full propagation profile, including peak timing, peak level, final reach, and the maximum growth rate, under the same parameter setting. In addition, the global sensitivity analysis identifies the exit-from-spreading and adoption mechanisms as the most influential levers overall, while the competition and conversion mechanisms between the two propagator types become clearly visible when type-specific trajectories are analyzed, especially for the second propagator class.

7.2 Implications for intervention strategies

The global sensitivity analysis clarifies which levers matter most for governance. Across the type-specific outcomes, the recovery or “exit-from-spreading” mechanism is the most influential factor, suggesting that interventions should prioritize accelerating clarification and correction processes, such as rapid fact-checking, timely official responses, and platform actions that reduce continued engagement with misleading content. The infection or adoption mechanism is the next key lever, which supports friction-based measures like “think-before-you-share” prompts, warning labels, and repost delays to reduce impulsive forwarding.

Importantly, using the two spreading types makes the competitive and conversion dynamics visible: the parameters governing competition between the two spreader types meaningfully affect the second type’s peak and final level. This implies that governance should not only suppress misinformation but also strengthen credible counter-narratives, for example, by boosting authoritative accounts and amplifying verified information to counter emotional narratives. Overall, effective strategies combine faster “exit” from spreading, reduced adoption of questionable content, and targeted shaping of the competition between credible and emotional narratives.

7.3 Limitations and future extensions

This paper mainly focuses on the theoretical analysis of the Hyper-S2IR model, including the derivation of the basic reproduction number R₀ and the proofs of local and global stability, with numerical simulations used to illustrate the corresponding dynamical behaviors. A first limitation is that spreader heterogeneity is represented by only two propagator types, HI₁ and HI₂. This stylized dichotomy is adopted for analytical tractability and to isolate the impact of heterogeneity on thresholds and stability, but real public sentiment may involve a broader and even continuous spectrum of stances and behaviors.

A second limitation is that the present study emphasizes mechanism and theory rather than case-specific empirical deployment. While the hypergraph formulation is compatible with constructing hyperedges from group-level interactions and the two propagator types admit behavior-based interpretations, a detailed operational mapping from platform records to hypergraph structure and propagator labels is left for future work. Likewise, systematic parameter calibration and benchmark-style comparisons under real datasets are beyond the scope of the current theoretical contribution.

A third limitation concerns scalability. Our simulations rely on a degree-stratified mean-field implementation that aggregates nodes into hyperdegree classes, and the computational behavior and approximation accuracy for very large-scale networks are not analyzed. Future work will investigate scalable stratification and coarse-graining strategies driven by hyperdegree and hyperedge-size distributions to retain key higher-order effects while improving computational feasibility.

Finally, the framework can be extended to Hyper-SnIR with more propagator classes and can be combined with Multi-Modal Hypergraph Transformer [34] and transformer-based deep learning [35] to support dynamic relation updating and finer interaction modeling, improving applicability to realistic public opinion settings.

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