With the persistent exploration of the information propagation mechanism, researchers have discovered the influencing factors of different individual behaviors on information outbreak [1–9]. Individual behavior is mainly affected by individual psychology. For example, in information propagation, some people show positive adoption, while others show negative adoption [10, 11]. These behaviors are not only applied in information propagation but also in other fields, such as economics, ecology, and medicine. By investigating abundant literature works, we found that there is a latent behavioral phenomenon in multiple fields. Particularly, in investment, when the investment quota is increased, the additional income brought by an increase in one unit of investment will decrease [12–15]. In agricultural production, when the number of chemical fertilizers increases, the additional yield that can be brought by a unit of chemical fertilizers will decrease [16, 17]. In medicine, when the dose of drugs increases, the treatment effect of drugs on patients will decrease [18, 19]. In the field of education, the additional educational effects that can be brought about by one unit of educational resources will be reduced [20, 21]. In summary, this behavior is called a decreasing behavior which is applied in multiple fields. For propagation dynamics, the paper aims to explore the impact mode of decreasing behavior on information propagation.

Based on our survey, we found that the phenomenon of decreased behaviors also exists in information propagation. For example, for advertising effect, at the beginning, advertising may attract people’s attention, but over time, the interest degree of people in advertising will gradually decrease [22, 23]. For media marketing, people’s interest degree in the same type of content will also decrease [24, 25]. For teaching, learning motivation of students for the same knowledge will decrease [26, 27]. In summary, in information propagation, if the same information is repeatedly received, the interest degree of people will decrease, which is closely related to individual psychology [28]. Therefore, this paper defines the phenomenon as an individual heterogeneous decreasing behavior, which is called IHDB, to explore its impact on information propagation mechanisms.

In this study, we found that people can only closely contact with a small number of friends due to the differences in society stratum, personal profession, and social acceptability [29–31]. For instance, a user can face difficulty to connect with all of his/her friends in a short period of time. Furthermore, several social media platforms, such as Twitter, Instagram, Tinder, and WhatsApp, are used frequently by many individuals [32–34]. Therefore, this paper considers the two-layer contacted network to explore the individual contact capacity which needs to be considered to analyze the information propagation mechanism.

More importantly, information propagation will be influenced by the individual adoption behaviors [35, 36]. According to the explanation given in Section 1.1, we found that the individual interest in behavior adoption is closely related to the proportion of neighbors who adopt the behavior. The individual heterogeneous decreasing behavior (IHDB) illustrates that when more and more neighbors transmit a same piece of information to the individual, the passion degree of the individual to receive the information gradually decreases.

The advantages of the IHDB compared to other behaviors are as follows: i) More accurate simulation of reality: IHDB reflects different responses and response degrees of the individual in information acceptance. This is closer to the real situation because the response of a person is gradually weakened in each receiving information, which will affect the information propagation process. ii) More accurate prediction of information propagation: By considering IHDB, you can more accurately predict the spread trend of information in the networks. With the proportion of neighbors in adoption state changes, the individual behavior spreads more passively or slowly. This difference can be reflected in the model, thereby providing more accurate diffusion forecasts. iii) Planning more targeted propagation strategies: The consideration of IHDB can help decision-makers formulate more targeted information dissemination strategies. By understanding the mental characteristics and behaviors of people, we can make better customized strategies for different types of information and improve the effect of information propagation. iv) Research and identification of crucial factors: Simulation of IHDB can help researchers identify the impact factors. By observing the IHDB in the individual, it is possible to determine the factors that play a key role in the information dissemination network so as to intervene or use these factors to promote information propagation.

Enlightened by the aforementioned overview, we consider individual limited contact capacity, capture the IHDB, and define a non-rule trapezoidal-like threshold function to illustrate the IHDB feature. The non-rule trapezoidal-like threshold function displays a slow and non-linear rising, subsequently maintaining a horizontal line. In addition, we provide a partition theory to analyze the information propagation mechanism based on limited contact and IHDB. Finally, through the theoretical analysis and experimental simulation, this paper reveals the information propagation mechanism.

The remainder of this paper is organized as follows: in Section 2, we put forward a probability adoption threshold model for information propagation on the two-layer contacted network. Section 3 exhibits a division theory based on limited contact and IHDB function model. Section 4 uses simulations and theoretical assessments to validate the information propagation mechanism. Finally, Section 5 summarizes the results, and Section 6 reports the conclusion.

A two-layered network model with N nodes is set up to explore the impacts of individual contact capability and IHDB characteristics on information propagation. Layers A and B represent two independent social network layers. Since the nodes connecting the various tiered networks are one-to-one correspondences, the same node is present in each layer. Then, the degree vector of node i is represented by k i ⃗ = ( k i A , k i B ) , where k i X represents a node degree. The degree distribution P ( k ⃗ ) of the network is represented by the degree vector k ⃗ . Furthermore, the degree distributions P _X (k ^X ) of P ( k ⃗ ) can be broken down in accordance with the uncorrelated feature. Considering the independence of P _A (k ^A ) and P _B (k ^B ), P ( k ⃗ ) = P A ( k A ) P B ( k B ) .

This paper establishes a two-layer social network model to explore a novel information propagation mechanism by exploiting S–A–R (susceptible–adopted–recovered) propagation theory, as illustrated in Figure 1A. The explanation of the S–A–R model theory is as follows: a node in the susceptible state is denoted as the S-state, in the adopted state is denoted as the A-state, and in the recovered state is denoted as the R-state. Specifically, the individuals in the S-state cannot propagate information, but they can receive it from their A-state neighbors. The A-state individuals have adopted a piece of information and will transmit it to their neighbors in the S-state. The individuals in the R-state no longer receive information and exit the whole propagation process. If an individual transforms its state in the X-layer in the two-layer network model, it will change to the same state in other layers.

This paper first considers that the contact ability of individuals can be set as C. If C < k _i , where k _i is the degree of individual i, an individual that has been adopted who can only contact some of its neighbors. However, it can contact all of its neighbors if C ≥ k _i . The probability of information transmission in the network can be set as λ. The probability that the information is received by the neighbor j in the S-state can be denoted as λ C k j X (X ∈ {A, B}).

m ^X is the total amount of information that the individual in the S-state has successfully received in the layer X. At the beginning, there is no information spreading on the multiple-layered network, the reason why the j value m j X of the individual in the S-state is 0. When a neighbor i in the A-state effectively spreads information to individual j along the related link in the A-layer or B-layer, the total information bits of individual j will rise by 1 at each time step, i.e., m j X → m j X + 1 . To explain the IHDB on information propagation, a non-rule trapezoidal-like function is proposed to illustrate the individual behavioral adoption in the whole network: h X x , α , β = x α β , 0 ≤ x < α , 0 < β < 1 1 , α ≤ x < 1 , (1) where x represents the proportion between the receiving information of an individual and the number of its contacted neighbors and parameter α represents the IHDB variable. In region I of Figure 1B, the behavioral adoption probability rises slowly to 1 based on the increase in x. In region II of Figure 1B, the adoption probability remains at 1.

We randomly select a portion of ρ ₀ individuals to act as the individuals (seeds) in the A-state at the beginning of information propagation and all other individuals to act as S-state individuals. Each A-state individual randomly chooses C of its neighbors to whom information is transferred in layer A(B) with probability λ ^A (λ ^B ). The total amount of information that individual j in the S-state effectively gets from layer X is m j X → m j X + 1 . Due to non-redundancy in information propagation, the information will not then be repeated through the same edge. Additionally, the individual in the susceptible state of X will accept the information and transmit to the adoption state with possibility h X ( m X k X , α , β ) at every time step. The S-state individual then changes to the corresponding state in other layers. Following successful information spreading, when the individual lost interest in information, the individuals in the adoption state transfer disinterested information and change to the recovery state with probability γ. In the two-layered contacted network, once there is no individual in the A-state, the information spreading process eventually comes to an end.

In Figure 2, with the unit transmission probability λ and the adoption threshold parameters α and β, we first investigate how the proportion of individuals in the three states of S-state, A-state, and R-state has changed with time.

Then, as shown in Figures 2A, B, R(t), which represents the ultimate spreading size, changes to 1 at the end, while S(t) increasingly decreases from 1 to 0 and A(t) progressively drops to 0 over time. As the increase in parameter C in subgraph (a) to subgraph (b), the evolution time costed steadily decreases from 9 to 6, while R(∞), which denotes the ultimate spreading size, increases at the same step t. The time progress demonstrates that information outbreak on a two-layered contacted network can be accelerated by enhancing contact ability of individuals.

Figure 3 displays the roles of unit transmission probability in each individual’s eventual adoption size with various IHDB values α in the subgraphs. The ultimate spreading size R(∞) spreads to a global network as λ rises, as shown in Figure 3A (β = 0.5) and (b) (β = 0.9). Furthermore, Figures 3A, B also indicate how IHDB can affect the propagation phase transition. In subgraph (a), the pattern of R(∞) always shows a second-order growth of continuous phase transition for any IHDB behavior exhibited by the individual (α ^A = α ^B = 0.1, 0.5, 0.9). The pattern of R(∞) in subgraph (b) indicates a second-order phase transition in the continuous propagation pattern when an individual exhibits a positive IHDB such as α ^A = α ^B = 0.1. This suggests that, when there is a small λ, a positive IHDB can result in widespread behavioral propagation. When α ^A = α ^B = 0.5, R(∞) also shows the same propagation phenomenon. While an individual exhibits a weak IHDB, R(∞) exhibits a first-order increase in the discontinuous pattern, i.e., α ^A = α ^B = 0.9.

Figure 3C displays the relative variances and critical information propagation probability of (a) and (b) individually (d). The global adoption will emerge from the deviation of behavioral propagation, which is represented by the top values of relative variance χ. Additionally, the numerical values of the simulation (symbols) agree with our theoretical analyses (lines).

For the two-layer contacted ER network, Figure 4 exhibits the joint impacts of variable (λ, α) on R(∞). Figure 4 (a) with β ^A = β ^B = 0.5 and (b) with β ^A = β ^B = 0.9 depict the effects of (λ, α) on the dissemination of information. In subgraph (a), the phase transition shows a continuous pattern in the whole area. Then, in subgraph (b), the image can be divided into two parts. In area I, a second-order continuous increase can be seen in the R(∞) pattern. The critical value between region I and region II is α* = 0.61. In area II, a first-order discontinuous increase can be seen in the R(∞) pattern. Additionally, the individual contact capability parameter is set at C = 5.

The joint impacts of variable plane (λ, β) on R(∞) for the two-layered ER network are shown in Figure 5. Figures 5A, B depict the effects of (λ, β) on information propagation, respectively, with α ^A = α ^B = 0.5 and α ^A = α ^B = 0.9. In subgraph (a), the image can be divided into two parts. There is a second-order continuous increase pattern in region I of R(∞) phase transition. The critical value between region I and region II is β* = 0.95. There is a first-order discontinuous increase pattern in region II of R(∞). Then, in subgraph (b), the image can also be divided into two parts. In area I, R(∞) exhibits a second-order continuous spreading. The critical value between region I and region II is β* = 0.78. In area II, R(∞) exhibits a first-order discontinuous spreading. Additionally, the individual contact capability parameter is set at C = 5.

Figure 6 depicts the influence of IHDB variable α and transmission probability λ on the ultimate spreading size of the multi-layer contacted SF network. In each subgraph, the fundamental parameters include C = 5 and β ^A = β ^B = 0.9. Figures 6A, B show how the ultimate adoption size R(∞) grows as λ increases until it achieves global adoption. When α ^A = α ^B = 0.1 and α ^A = α ^B = 0.5, the final spreading size shows a continuous spreading with second-order. However, R(∞) pattern exhibits a discontinuous spreading with first-order when α ^A = α ^B = 0.9. Then, the same growth pattern is also exhibited in subgraph (b) (v = 4). Moreover, compared with subgraph (b) (v = 4), subgraph (a) (v = 2) shows an incomplete global adoption because of strong heterogeneous degree distribution. Additionally, the numerical values of the simulation (symbols) match those of our theoretical analyses (lines).

For the multiple-layered contacted SF network, Figure 7 shows the effect of R(∞) on the behavioral parameter plane (λ, β). The subgraphs (a) and (b), and (c) and (d) are set as the identical contact capacity of individuals by C = 5 and C = 10, respectively. The subgraphs (a) and (b), and (c) and (d) demonstrate the growth tendency of R(∞). In subgraph (a) with v = 2 and C = 5, the image can be divided into three parts. In the phenomenon of eventual information outbreak R(∞), area I displays a second-order continuous propagation pattern. The critical value between region I and region II is β* = 0.85. Area II exhibits a first-order discontinuous phase transition in the pattern of R(∞). The critical value between region II and region III is β** = 0.99. In area III, the R(∞) pattern does not exhibit information outbreak. In subgraph (b) with v = 2 and C = 10, the image can be divided into two parts. The R(∞) pattern of individual final spreading size in area I displays a second-order continuous increase. The critical value between region I and region II is β* = 0.75. In the pattern of R(∞), area II displays a first-order discontinuous increase. In subgraph (c) with v = 4 and C = 5, the image can be divided into two parts. While showing the pattern of information outbreak scale R(∞), area I displays a second-order continuous propagation pattern. The critical value between region I and region II is β* = 0.8. A first-order discontinuous increase in area II’s pattern of R(∞) is visible. In subgraph (d) with v = 4 and C = 10, the image can be divided into two parts. In the pattern of individual eventual spreading size R(∞), area I displays a second-order continuous increase. The critical value between region I and region II is β* = 0.85. Area II exhibits a first-order discontinuous phase transition in the pattern of R(∞). Furthermore, the heterogeneous degree distribution alters the information propagation but cannot alter the pattern of phase transition. Because there are some hub people in the multi-layer contacted SF network, when it exhibits a strong heterogeneous degree distribution (v = 2), there is a pattern of information suppression in the phase transition.

Figure 8 exhibits the impact of IHDB variable α and transmission probability λ on the final spreading scope for the two-layer contacted ER–SF network. In each subgraph, the fundamental parameters include C = 5 and β ^A = β ^B = 0.9. Figures 8A, B (v ^B = 2 and v ^B = 4) show how the final spreading scope R(∞) grows as λ increases until it achieves global adoption. When α ^A = α ^B = 0.1 and α ^A = α ^B = 0.5, the final outbreak pattern shows a second-order continuous propagation. However, the R(∞) pattern shows a first-order discontinuous pattern when α ^A = α ^B = 0.9. In addition, the numerical values of the simulation (symbols) match those of our theoretical analyses (lines).

For the two-layer contacted ER–SF network, Figure 9 shows the effect of R(∞) on the behavioral parameter plane (λ, β). The subgraphs (a) and (b) are set as the identical contact capacity of individuals by C = 5. The subgraphs (a) and (b) demonstrate the growth tendency of R(∞). In subgraph (a) with v = 2, the image can be divided into two parts. In the phenomenon of eventual information outbreak R(∞), area I displays a second-order continuous propagation pattern. The critical value between area I and II is β* = 0.86. Area II exhibits a first-order discontinuous phase transition in the pattern of R(∞). In subgraph (b) with v = 4, the image can be divided into two parts. The R(∞) pattern of individual final spreading size in area I displays a second-order continuous increase. The critical value between region I and region II is β* = 0.9. In the pattern of R(∞), area II displays a first-order discontinuous increase.