In the field of network science, the study of social communication has garnered widespread attention among researchers [1, 2]. It can be applied to analyze financial behaviors [3, 4], social information diffusion [5], and emotional contagion [6]. Furthermore, it can be utilized for disaster prediction [7] and risk mitigation [8]. Scholars have explored the mechanisms of social communication through both theoretical analysis and extensive experimental validation. Research has revealed that social communication exhibits certain unique reinforcement effects compared to biological propagation [9, 10].

In the early stages of research, the most commonly used approach was the threshold model based on memoryless Markov processes [11]. In this threshold model, a behavior is adopted when the number of adopting neighbors exceeds a predetermined threshold [12, 13]. Given the small proportion of initial seeds, the initial infection rate is predicted using percolation principles [14, 15]. Based on the assumption of constant thresholds, because of variations in the average degree, saddle-node bifurcation occurs, leading to a continuous increase and subsequent discontinuous decrease in the final adoption size with increasing network average degree. Research has found that factors such as the initial number of seeds [16, 17], clustering coefficient [18], multilayer networks [19], network temporal dynamics [20] and time-varying [21] significantly affect information propagation in the threshold model.

In the field of complex networks, numerous scholars have conducted extensive research on propagation behaviors in single-layer networks. However, studies have shown that multilayer networks better represent real-world social networks. In the context of bio-information networks [22], individuals can access various information in the information network to execute different strategies in the biological network [23]. For example, during the COVID-19 pandemic, people could obtain preventive measures through the internet, leading to improved habits and reduced chances of contracting the virus in the biological network. In the case of multilayer information networks, individuals often do not rely on a single channel to interact with the external world. Each person has multiple social channels, such as WeChat, Twitter, Instagram, and more. Thus, in a multilayer network, information does not propagate solely within a single layer but rather disseminates through multiple coupled networks. However, individual information acquisition remains singular [24]. For instance, on the YouTube platform, users can upload and share video content, and other users can subscribe to their channels. This forms a user-user connectivity network. Each video can be viewed, commented on, and shared by other users, creating a video-user connectivity network. When a user uploads a video, their subscribers can see it in their subscription feed. If these subscribers find the video appealing or valuable, they can choose to share it with their own audience. Consequently, the video spreads through user-to-user sharing in the network. In this example, the social network among users and the video-user connectivity network constitute a multilayer network. Video content propagates through sharing and viewing behaviors among users, and this multilayer network structure can influence the dissemination path, view counts, and impact of videos. In conclusion, multilayer networks better capture the essence of real-world networks, allowing us to simulate human behavior in real-life situations.

With the diversification of information channels and the complexity of information forms, people are living in an era of fragmented information, and the adoption patterns of information have gradually become differentiated and heterogeneous. This is especially true for differentiating information sources. For information channels with low trustworthiness, people may encounter a mix of correct and incorrect information, leading to skepticism towards the information from these channels. Conversely, for information channels with high trustworthiness, such as those associated with authoritative sources or long-standing trust, people are more likely to trust and adopt information from these channels [25]. Therefore, the two-layer network model [26] takes into account the more complex information transmission, which has significant practical implications. Previous research has not extensively addressed heterogeneous threshold functions. While some studies have explored two-layer networks, they often assume the same threshold function for both layers. However, in real life, due to the heterogeneity between layers, people have different levels of trust and acceptance for information from different sources. Consequently, people behave differently after acquiring information from different platforms. Therefore, adopting heterogeneous functions for different channels better reflects reality. In this study, a heterogeneous threshold function is employed in the two-layer model to capture this phenomenon.

Existing research has shown that individuals exhibit different adoption attitudes towards the same information on different network layers, and their attitudes may change as the amount of information they receive fluctuates [27]. However, there is relatively limited research on considering heterogeneous adoption in information propagation within complex networks. In real social networks, individuals vary in their level of adoption across different layers of information. Based on the adoption attitudes towards information on different layers, this study categorizes heterogeneous adoption in social networks into two types: regular adoption and hesitant adoption. Regular adoption refers to a linear increase in the willingness to adopt with the number of received information or behaviors. Hesitant adoption, on the other hand, involves a state of hesitation regarding whether to adopt, requiring repeated verification of information and accumulating more information before developing the willingness to adopt. For instance, when a popular piece of information appears on the internet, an individual is more likely to increase their trust and adopt it on reliable information platforms, leading to a rapid saturation of adoption on such platforms. However, when the same individual encounters this information on an untrustworthy platform, they may exhibit a hesitant adoption stance, repeatedly verifying the information before deciding to adopt. As a result, the propagation speed on such platforms is slower, and it takes some time for the adoption to reach a relative saturation point. Therefore, studying the behavioral division of inter-layer adoption heterogeneity will contribute to a deeper understanding of the propagation mechanisms in multi-layer social networks.

The paper proposes a heterogeneous threshold adoption function on a two-layer model and constructs a heterogeneous adoption behavior network model for the same information on the two layers. It investigates the heterogeneous information propagation in a heterogeneous network of the same population. Through extensive simulation and theoretical analysis, the study reveals that when either layer dominates, the final outbreak of the adoption range manifests a second-order continuous phase transition. In contrast, when there is no clear dominant layer, the outbreak follows a first-order discontinuous phase transition. The timing and extent of the outbreak are influenced by various factors such as hesitant adoption parameter, degree heterogeneity parameter, and propagation probability. In the steady state, the final propagation range reaches global dissemination.

Based on Ref. [28], this study employs an edge-based compartmental (EBC) method for theoretical analysis of the model. By analyzing the variation in the number of individuals in different states in a multilayer network, the study provides a theoretical evaluation of the propagation mechanism in a heterogeneous adoption behavior network for the same information in a dual-layer setting.

Inspired by the “Hole Theory” [29], it shows that individual i is in a “hole” state, meaning it cannot transmit information to its neighbors but can receive information from them. Let θ k j X X ( t ) ( X ∈ A , B ) represent the probability that an individual with degree k _j has not transmitted information to i until time t. Then, in different layers, the probability that individual i has not received any information until time t can be expressed as Eqs 3, 4: θ A t = ∑ k j A = 0 k j A P k j A k A θ k j A A t (3) and θ B t = ∑ k j B = 0 k j B P k j B k B θ k j B B t (4)

Based on the assumption that k j X P ( k j X ) k X ( X ∈ A , B ) represents the probability of node j being connected to node i in layer X, it can be derived that the probability that node i accumulates m _X non-redundant information in layer X at time t as Eq. 5: ϕ m X X k i X , m X , t = k i X m X θ X t k i X − m X 1 − θ X t m X (5)

Therefore, the probability that node i does not adopt any behavior in layer A and remains in state S is ∏ l = 0 m A [ 1 − h a ( x , α , β ) ] , and the probability that node i does not adopt any behavior in layer B and remains in state S is ∏ l = 0 m B [ 1 − h b ( x , α ) ] . It can be then calculated that the probability that node i remains in state S after receiving m _X non-redundant information in both layer A and layer B until time t as Eqs 6, 7: τ A k i A , m A , t , α , β = ∑ r = 0 m ϕ m A A k i A , t ∏ l = 0 r 1 − h a k i A l , α , β = ∑ r = 0 α k i A ϕ m A A k i A , t ∏ l = 0 r 1 − β − 0.5 ⁡ cos 2 π k i A α l + 0.5 + ∑ n = a k i A m ϕ m A A k i A , t ∏ l = 0 α k i 1 − β − 0.5 ⁡ cos 2 π k i A α l + 0.5 ∏ l = a k i A n 1 − − 0.5 ⁡ cos π x − α 1 − α + 0.5 (6) τ B k i B , m B , t , α , β = ∑ r = 0 m ϕ m B B k i B , t ∏ l = 0 r 1 − h b k i B l , α = ∑ r = 0 1 − α k i B ϕ m B B k i B , t ∏ l = 0 r 1 − − 0.5 ⁡ cos π k i B 1 − α l + 0.5 + ∑ n = a k i B m ϕ m B B k i B , t ∏ l = 0 1 − α k i B 1 − − 0.5 ⁡ cos π k i B 1 − α l + 0.5 ∏ l = a k i B n 1 − 1 = ∑ r = 0 1 − α k i B ϕ m B B k i B , t ∏ l = 0 r 0.5 + 0.5 ⁡ cos π k i B 1 − α l (7)

So, the probability that node i, with k ⇀ i = ( k i A , k i B ) , remains in state S after accumulating m _A and mB messages in networks A and B respectively until time t can be calculated as Eq. 8: s ( k ⇀ , t ) = 1 − ρ 0 ∑ m A = 0 k i A ϕ m A A k i A , t ∏ l = 0 m A 1 − h a k i A l , α , β × ∑ m B = 0 k i B ϕ m B B k i B , t ∏ l = 0 m B 1 − h b k i B l , α = 1 − ρ 0 τ A k i A , m A , t , α , β τ B k i B , m B , t , α , β (8)

If S(t), A(t), R(t) are used to represent the proportions of nodes in different states, considering nodes with different degrees, the proportion of nodes in the susceptible state at time t can be expressed as Eq. 9: S t = ∑ k ⇀ P X k ⇀ s k ⇀ , t (9)

Considering that the neighbors of individual i can be in the susceptible, adopter, or recovered state, θ k j X X ( t ) can be further expressed as Eq. 10: θ k j X X t = ξ S , k j X X t + ξ A , k j X X t + ξ R , k j X X t (10)

In this case, ξ S , k j X X ( t ) , x i A , k j X X ( t ) , x i R , k j X X ( t ) represent the probabilities of neighbor node j being in the susceptible, adopter, and recovered states, respectively, and not having transmitted information to node i until time t. Since node i is in a “hole” state, it cannot transmit information to node j. Therefore, the probability that node j accumulates n _X non-redundant information in layer X at time t can be expressed as Eqs 11, 12: ζ A k j A − 1 , n A , t = ∑ n A = 0 k j A − 1 ϕ n A A k j A − 1 , n A , t ∏ l = 0 n A 1 − h a x , α , β (11) ζ B k j B − 1 , n B , t = ∑ n X = 0 k j B − 1 ϕ n B B k j B − 1 , n B , t ∏ l = 0 n B 1 − h b x , α (12)

In layer X, the probability that node j remains in the susceptible state at time t is as Eqs 13, 14: ξ S , k j X A t = 1 − ρ 0 1 − 1 − ζ A k j A − 1 , n A , t 1 − τ B k j B , n B , t (13) and ξ S , k j X B t = 1 − ρ 0 1 − 1 − ζ B k j B − 1 , n B , t 1 − τ A k j A , n A , t (14)

Since the probability of transmitting information through edges is λ, and the recovery probability of adopter nodes is γ, the equation for ξ R , k j X X ( t ) as Eq. 15: d ξ R , k j X X t d t = γ 1 − λ ξ A , k j X X t (15)

At time t, the probability of information being transmitted through an edge is equal to the probability of an adopter node transmitting the information to a susceptible neighbor. Therefore, Eq. 16 is as follows: d θ k j X X t d t = − λ ξ A , k j X X t (16)

By combining Eqs 12 and 13, Eq. 17 can be obtained: ξ R , k j X X t = γ 1 − λ 1 − θ k j X X t λ (17)

By substituting Eqs 8 and 14 into Eq. 13, Eq. 18 can be obtained: d θ k j X X t d t = − λ θ k j X X t − ξ S , k j X X t + γ 1 − λ 1 − θ k j X X t (18)

Given the initial conditions for θ _X (0) = 1 and ξ R , k j X X ( t ) = 0 , when t → ∞ the Eq. 15 equals 0, Eq. 19 can be derived the expression for θ k j X X ( t ) as: θ k j X X t = λ ξ S , k j X X t + γ 1 − λ γ 1 − λ + λ (19)

By substituting the expression for θ k j X X ( t ) into Eqs 2 and 3, Eq. 20 can be obtained: θ X ∞ = ∑ k j X = 0 k j X P k j X k X θ k j X X ∞ = f X θ A ∞ , θ B ∞ (20)

To simplify the notation, let’s use the function f(x) to represent θ _X (∞).

By substituting the obtained equations into (4)–(7), it can be derived that the proportion of susceptible nodes S (∞). Since the growth of d A ( t ) d t is due to the decrease in S(t), so the equations for the proportions of nodes in different states as Eqs 21, 22: d A t d t = − d S t d t − γ A t (21) and d R t d t = γ A t (22)

Based on S (∞), it can be obtained that R (∞) as a complement to 1, since the proportions of nodes in all states must sum up to 1.

To further investigate the conditions for non-continuous growth of the function, the situation can be determined when Eq. 17 is tangent to θ _X (∞) < 1 by calculating the following Eq. 23: ∂ f A θ A ∞ , θ B ∞ ∂ θ B ∞ ∂ f B θ A ∞ , θ B ∞ ∂ θ A ∞ = 1 (23)

To ensure simulation accuracy, a minimum of 10³ dynamic realizations are recommended in the network for this study. The network size is set to N = 10⁴, with an average degree of k A = k B = k 10 = 10 . To investigate the impact of contact capacity on information propagation mechanisms in ER-ER and SF-SF networks, the network layer X X ∈ A , B in ER-ER network follows a Poisson degree distribution p X ( k X ) = e − k X k X k X k X ! , while in SF-SF network, the network layer follows a power-law degree distribution p X ( k X ) = ξ X k X − v . Here, ξ X = 1 ∑ k X k X − v and ν are parameters representing the degree exponent of layer A and layer B, respectively.

The heterogeneity of the network degree distribution is negatively correlated with the degree distribution exponent ν. When ν is small, the network contains a few high-degree nodes and many low-degree nodes. Additionally, to make the process more convenient, the information transmission probability is set as λ _A = λ _B = λ, and the recovery rate is set as γ = 1.0.

The peak of the relative variance χ curve of the final adoption range and the corresponding information transmission probability at the critical points are as Eq. 24: χ = N ⟨ R ∞ 2 ⟩ − R ∞ 2 R ∞ (24)

The symbol … here represents the ensemble average.

In this paper, the first exploration is about the propagation of information on a weighted ER network, where the nodes in the ER network follow a Poisson distribution, denoted as [ P k = e − k k k / k ! . The simulation results represented by symbols and the predicted results represented by lines show consistent trends.

In real-life, individuals exhibit different social behaviors within various social networks. To analyze the propagation mechanisms and investigate heterogeneous adoption behavior in a heterogeneous network of the same population, this study proposes a dual-layer heterogeneous adoption information propagation network model from both simulation and theoretical perspectives. A heterogeneous threshold function based on realistic psychological research is designed, and extensive experiments demonstrate the consistent results between simulations and theory.

This paper focuses on the innovative aspect of heterogeneous adoption behavior within a dual-layer model and explores the propagation of heterogeneous behavior within the same population in a dual-layer heterogeneous network. Through extensive simulation and theoretical analysis in SF and ER networks, it is observed that when either layer dominates, the final adoption range exhibits a second-order continuous phase transition. In the absence of a clear dominant layer, a first-order discontinuous phase transition occurs with the presence of cross-propagation phenomena. The propagation process and modes are influenced by factors such as hesitation parameters, degree heterogeneity parameters, and propagation probabilities, ultimately leading to complete propagation.

The inter-layer adoption heterogeneity in information propagation networks has a crucial impact, yet there is limited research in this area. This paper rigorously models and analyzes the significant influence of heterogeneous adoption behavior within multi-layer networks on information propagation. The study also provides a new direction for information propagation in multi-layer heterogeneous networks. However, this article has certain limitations. Firstly, it does not use real datasets, lacks standardized data representing human behavioral characteristics, and cannot extensively validate the behavior propagation with real-world data. Secondly, it does not consider several conventional parameters that may influence the research process, such as weights and fluctuation-based adoption. To emphasize the influence of heterogeneous adoption behavior within this model, other parameters were reduced to better highlight the significance of studying heterogeneous adoption behavior. It is hoped that more experts and scholars will pay attention to this field and further expand the research.