# Effect of Contact Preference among Heterogeneous Individuals on Social Contagions

Authors: Yining Xu; Jinghua Xiao; Xiaochen Wang
Journal: Complexity
 (2022)

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## Abstract

In social networks, individual heterogeneity is widely existed, and an individual often tends to contact more frequently with friends of similar status or opinion. It is worth noting that the contact preference characteristic among heterogeneous individuals will have a significant effect on social contagions. Thus, we propose a social contagion model which takes the heterogeneity of individual influence and contact preference into account, and make a theoretical analysis of the social spreading process by developing an edge-based compartmental theory. We find that the competition between simple contagion and complex contagion leads to the emergence of crossover phase transition phenomena when the influence of ordinary individuals is low: it changes from being hybrid to continuous, then to hybrid, and eventually to continuous phase transition with the increase of the contact probability of homogeneous individuals. However, it changes from being first-order to continuous phase transition when the influence strength of ordinary individuals is relatively high. In addition, there is an optimal value or range of contact probability of homogeneous individuals which maximizes the spreading size, and the optimal value or lower bound of the optimal range decreases with the increase of transmission probability. Our theoretical prediction results are in good agreement with the simulation results.


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## Body

## 1. Introduction

In real life, the influence or reputation of individuals is often heterogeneously distributed. Some individuals are more influential than others and the scholars called them “opinion leaders” [1]. Rogers [2] indicated that the innovators always firstly adopt the new technologies in the early stage of innovation diffusion and they play the role of “opinion leaders.” It is worth noting that the contact preference among individuals with heterogeneous influence strength will seriously affect the process of social contagion, especially for the spreading of new concepts or products with high adoption risk or cost. For example, since it is often difficult for ordinary individuals to contact or get familiar with medical staffs with authoritative background, most individuals cannot gain an in-depth understanding of the COVID-19 vaccine in the early stage of its promotion and did not make proactive vaccinations considering the risk of vaccination and the uncertainty of immunization effects which delays the containment process of the COVID-19 spreading [3]. Therefore, it is critical to study the influence of contact preference among heterogeneous individuals on spreading processes, which can help us to develop more effective strategies to popularize new concepts and knowledge or contain epidemic spreading.A large number of experimental and theoretical results show that social contagion is different from disease contagion in which each contact has an independent probability of infection. In social contagion, a person will refer to his/her friends' opinion before adopting a piece of information [4–8]. He/she is more likely to be affected as more and more friends adopt it, which is the so-called social reinforcement. So far, the modeling and analysis of social contagion process have attracted a lot of attentions. As the adoption of behaviors or products often requires a certain cost and the reinforcement plays an important role, scholars often use threshold model [5, 9–11] to study social contagion. At present, the impact of individual heterogeneity on information diffusion, behavior spreading, and other social contagion processes has been widely concerned by scholars [12–15]. Scholars have studied the influence of heterogeneous degree distribution [16], adoption threshold [17, 18], and waiting time [19] on the social contagion processes. Studies show that heterogeneity can significantly affect the final spreading size and phase transition of social contagion [16, 20]. For example, Tang et al. found that the heterogeneous adoption threshold of individuals causes the final spreading size to change from being first-order phase transition to continuous transition when varying the transmission probability [17]. Then, Lee and Holme studied the effect of degree-related influence strength on social contagion in different networks and found that discontinuous transitions will occur if nodes with large degree value have greater influence strength [21]. Wang et al. considered the coupling between degree-related influence strength and adoption threshold and found that the final state of system will be a mix of different opinions when the coupling is strongly positive [22]. Moreover, Easley and Kleinberg have studied how heterogeneous individuals with different influence strength play a role in the spreading process and found the phenomena of crossover phase transitions by varying the distribution of individuals with different influence strength [23].In real spreading process, the contact among heterogeneous individuals is often preferential. For instance, Hill and Dunbar analyzed the behavior of exchanging Christmas cards in the West and found that individuals have a higher rate of contacting with people they are familiar with [24]. Brown and Enos measured individual partisan segregation by calculating the local residential segregation of every registered voter in the United States and presented evidence of extensive partisan segregation in the country [25]. Many studies have mentioned that individuals’ different interaction attributes in behavior and information spreading processes usually have a certain impact on behavior prediction [26, 27]. Merkley and Loewen demonstrate that intergroup contact between advantaged group and disadvantaged group can undermine support for social change towards greater equality by analyzing a large and heterogeneous dataset (12,997 individuals from 69 countries) [28]. Hssler et al. provided experimental evidence of anti-intellectualism’s importance in information search behavior and found anti-intellectualism (the generalized distrust of experts and intellectuals) poses a fundamental challenge in maintaining and increasing public compliance with expert-guided COVID-19 health directives [29]. Therefore, some scholars have studied the spreading process with contact preference. Cui et al. proposed the concept of intimate contact and ordinary contact to reflect the contact preference among individuals and found that intimate contact plays a decisive role in the outbreak of contagion, while ordinary contact plays a decisive role in large-scale contagion [30]. Later, Yang et al. studied the influence of preferential contact process on contagion dynamics in irrelevant networks; the results show that preferential selection of small degree node can greatly improve the spreading size [31]. For degree-dependent networks, Gao et al. proposed a preferential contact strategy based on local network structure and knowledge density to promote information spreading; the results show that a moderate correlation coefficient can result in the most efficient information spreading [32]. Moreover, Lin et al. proposed the concept of optimization index, which determines whether the propagation is more biased towards the nodes of large degree. Finally they found the phenomena of crossover phase transitions and the optimal index which makes the spreading size reach the maximum value [33].Individual heterogeneity widely existed in social contagions, and there is still a lack of relevant research on how the contact preference among heterogeneous individuals affects the process of social contagion, which is crucial for us to develop effective strategies to promote or control social contagions. Moreover, it is still unknown what kind of contact preference is most optimal to maximize the spreading size. Therefore, we propose a social contagion model considering the contact preference among heterogeneous individuals and theoretically analyze the spreading process with the help of an edge-based compartmental theory. We find the crossover phase transition phenomena when increasing the contact probability of homogeneous individuals. When the influence strength of ordinary individuals is low, an individual needs multiple reinforcements from ordinary individuals to be persuaded. In this case, we find the competition between simple contagion and complex contagion leads to the emergence of crossover phase transition phenomenon: it changes from being hybrid to continuous phase transition, then to hybrid, and eventually to continuous phase transition. However, we find another crossover phase transition phenomenon when the influence strength of ordinary individuals is relatively high and at this time the social reinforcement is weakened: it changes from being first-order to continuous phase transition. In addition, we explore the optimal contact probability of homogeneous individuals which allows the spreading size to reach the maximum value. There is an optimal value when the transmission probability is low. In this case, the value of optimal contact probability decreases with the increase of transmission probability. And there will be an optimal range of contact probability when the transmission probability is relatively large. Similarly, we find the lower bound of optimal range decreases with the increase of transmission probability.


## 2. System Model

### 2.1. Network Model

Considering the heterogeneous distribution of influence strength of individuals, we roughly divide individuals into two types, i.e., type-A individuals with general influence strengthIA and type-B individuals with relatively high influence strength IB (IA<IB). Type-A individuals represent the ordinary people without professional knowledge and have less influence in the process of information contagion, type-B individuals represent authoritative persons with professional knowledge and have a high persuasion in the process of information contagion. Among them, type-A and type-B individuals are randomly distributed in the social network. Their proportions are represented by ρA and ρB, respectively, and satisfy ρA+ρB=1.Based on the contact preference among heterogeneous individuals, we constructed a social network withN individuals. At each step of the network construction, two individuals which are not connected are randomly selected. Their contact probability is p if the two individuals are the same type; otherwise their contact probability is 1−p. There is no contact preference among heterogeneous individuals in the social network when p=0.5. Different types of individuals tend to be connected when p<0.5; i.e., type-B individuals with high influence strength tend to connect with type-A individuals. In the case of p>0.5, individuals of the same type are more likely to be connected; that is, type-B individuals and type-A individuals tend to connect with the same type of individuals. Given the degree value of each individual under certain degree distributions of type-A and type-B individuals, we can repeat the above connection steps until all individuals meet their own degree and the network construction process ends.


### 2.2. Social Contagion Model

Based on the constructed social network, we propose a threshold model to reflect the social contagion process with contact preference among heterogeneous individuals. In the social contagion model, individuals are divided into three groups according to their spreading states: susceptible state (S), adopted state (A), and recovered state (R). In addition, we also consider the spreading characteristics of nonredundant information; that is, individuals have a transmission memory and will not repeat the spreading process to the same neighbors.In the process of social contagion, each individual has the same adoption thresholdT for simplicity. At each time step, individuals in the adopted state spread information to susceptible neighbors with transmission probability λ. If a susceptible individual j is informed by an adopted neighbor i, the cumulative influence Infj he/she has received is increased by Ii (that is, Infj=Infj+Ii, Ii=IA (or IB) when i is a type-A (or type-B) individual). When the cumulative influence strength that the susceptible individual j received is no less than his/her own threshold (i.e., Infj≥T), the susceptible individual j becomes adopted. At the end of each time step, each adopted individual recovers with possibility γ. When there are no adopted individuals in the system, the spreading process ends.


## 2.1. Network Model

Considering the heterogeneous distribution of influence strength of individuals, we roughly divide individuals into two types, i.e., type-A individuals with general influence strengthIA and type-B individuals with relatively high influence strength IB (IA<IB). Type-A individuals represent the ordinary people without professional knowledge and have less influence in the process of information contagion, type-B individuals represent authoritative persons with professional knowledge and have a high persuasion in the process of information contagion. Among them, type-A and type-B individuals are randomly distributed in the social network. Their proportions are represented by ρA and ρB, respectively, and satisfy ρA+ρB=1.Based on the contact preference among heterogeneous individuals, we constructed a social network withN individuals. At each step of the network construction, two individuals which are not connected are randomly selected. Their contact probability is p if the two individuals are the same type; otherwise their contact probability is 1−p. There is no contact preference among heterogeneous individuals in the social network when p=0.5. Different types of individuals tend to be connected when p<0.5; i.e., type-B individuals with high influence strength tend to connect with type-A individuals. In the case of p>0.5, individuals of the same type are more likely to be connected; that is, type-B individuals and type-A individuals tend to connect with the same type of individuals. Given the degree value of each individual under certain degree distributions of type-A and type-B individuals, we can repeat the above connection steps until all individuals meet their own degree and the network construction process ends.


## 2.2. Social Contagion Model

Based on the constructed social network, we propose a threshold model to reflect the social contagion process with contact preference among heterogeneous individuals. In the social contagion model, individuals are divided into three groups according to their spreading states: susceptible state (S), adopted state (A), and recovered state (R). In addition, we also consider the spreading characteristics of nonredundant information; that is, individuals have a transmission memory and will not repeat the spreading process to the same neighbors.In the process of social contagion, each individual has the same adoption thresholdT for simplicity. At each time step, individuals in the adopted state spread information to susceptible neighbors with transmission probability λ. If a susceptible individual j is informed by an adopted neighbor i, the cumulative influence Infj he/she has received is increased by Ii (that is, Infj=Infj+Ii, Ii=IA (or IB) when i is a type-A (or type-B) individual). When the cumulative influence strength that the susceptible individual j received is no less than his/her own threshold (i.e., Infj≥T), the susceptible individual j becomes adopted. At the end of each time step, each adopted individual recovers with possibility γ. When there are no adopted individuals in the system, the spreading process ends.


## 3. Theoretical Analysis

### 3.1. Spreading Process Analysis

Since the state change of susceptible individuals depends on the influence strength distribution and spreading states of neighbors, we use the edge-based compartmental theory to analyze the social contagion process with contact preference among heterogeneous individuals. Assuming that the influence strength of type-B individuals isIB=T, it means that they can persuade their neighbors directly. The ratio of influence strength between type-A and type-B individuals is expressed by r0=IA/IB (0<r0<1).Due to the existence of contact preference among heterogeneous individuals, the type distributions of neighbors of type-A and type-B individuals are different. Therefore, we will analyze the adoption processes of type-A and type-B individuals, respectively. For a random edge,θAt is defined as the probability that a type-A neighbor iA of individual j does not spread information to j until time t. θBt is defined as the probability that a type-B neighbor iB of individual j does not spread information to j until time t. Suppose that j has k neighbors, including n type-B individuals, and receives m messages until time t. The probability that all messages come from his/her k−n type-A neighbors is(1)ϕmk,n,t=Ck−nmθAtk−n−m1−θAtmθBtn.Ifj belongs to type-A, the probability that a randomly selected neighbor of j belongs to type-B (or type-A) is pAB (or pAA), and they satisfy pAB+pAA=1. If j belongs to type-B, the probability that a randomly selected neighbor of j belongs to type-B (or type-A) is pBB (or pBA), and they satisfy pBB+pBA=1. By counting the number of edges between the same type of individuals, we can get pAA and pBB in a given networks. Let sAk,n,t (or sBk,n,t) denote the probability that type-A (or type-B) individuals have n type-B neighbors and maintain susceptible at time t. Each individual can get at most c times of information from his/her type-A neighbors to maintain susceptible state. If k−n<c, then(2)sAk,n,t=CknpABnpAAk−n∑m=0k−nϕmk,n,t,sBk,n,t=CknpBBnpBAk−n∑m=0k−nϕmk,n,t.Ifk−n≥c, then(3)sAk,n,t=CknpABnpAAk−n∑m=0cϕmk,n,t,sBk,n,t=CknpBBnpBAk−n∑m=0cϕmk,n,t.Considering all possible values ofn, sAk,t (or sBk,t) represents the probability that a type-A (or type-B) individual with degree k is susceptible at time t. If k<c, then(4)sAk,t=∑n=0kCknpABnpAAk−n∑m=0k−nϕmk,n,t,sBk,t=∑n=0k−nCknpBBnpBAk−n∑m=0k−nϕmk,n,t.Ifk≥c, then(5)sAk,t=∑n=0k−cCknpABnpAAk−n∑m=0cϕmk,n,t+∑n=k−c+1kCknpABnpAAk−n∑m=0k−nϕmk,n,t,sBk,t=∑n=0k−cCknpBBnpBAk−n∑m=0cϕmk,n,t+∑n=k−c+1kCknpBBnpBAk−n∑m=0k−nϕmk,n,t.In the initial stage, a vanishingly small proportionε0 of individuals are randomly selected as seeds, the degree distributions of type-A (or type-B) individuals satisfy PAk (or PBk), and the susceptible proportion of type-A and type-B individuals at time t is, respectively,(6)SAt=1−ε0∑k=kminkmaxPAksAk,t,(7)SBt=1−ε0∑k=kminkmaxPBksBk,t,where (1 −ε0) represents the proportion of susceptible individuals in the initial state. Then the proportion of susceptible individuals in the network at time t is(8)St=ρASAt+ρBSBt.According to the above analysis, we can getSAt, SBt, and St by computing θAt and θBt. Here we will analyze how to compute θAt and θBt.According to the edge-based compartmental theory, since neighborsiA and iB of j may have possible three states: susceptible, adopted, and recovered state, we can express θAt and θBt as follows:(9)θAt=ξASt+ξAAt+ξARt,(10)θBt=ξBSt+ξBAt+ξBRt,where ξASt [ξAAt or ξARt] denotes the probability that a type-A neighbor iA is in S state [A state or R state] at time t and does not transmit information to j, and ξBSt [ξBAt or ξBRt] denotes the probability that a type-B neighbor is iB in S state [A state or R state] at time t and does not transmit information to j.We first analyzeξASt and ξBSt. When iA (or iB) is susceptible with k′ neighbors, he/she can receive messages from his/her k′−1 neighbors (as j which is susceptible and cannot transmit message). Assuming that, among the k′−1 neighbors of iA (or iB), there are n type-B individuals and k′−1−n type-A individuals, then iA (or iB) receives m messages at time t, and the probability that all the m messages come from his/her k′−1−n type-A neighbors is(11)τmk′,n,t=Ck′−1−nmθAtk′−1−n−m1−θAtmθBtn.Considering all possible values ofn, the probability of iA (or iB) with degree k′ maintaining susceptible at time t is as follows: if k′−1<c, then(12)sA′k′,t=∑n=0k′−1Ck′−1npABnpAAk′−1−n∑m=0k′−1−nτmk′,n,t,sB′k′,t=∑n=0k′−1Ck′−1npBBnpBAk′−1−n∑m=0k′−1−nτmk′,n,t.Ifk′−1≥c, then(13)sA′k′,t=∑n=0k′−1−cCk′−1npABnpAAk′−1−n∑m=0cτmk′,n,t+∑n=k′−ck′−1Ck′−1npABnpAAk′−1−n∑m=0k′−1−nτmk′,n,t,s′k′,t=∑n=0k′−1−cCk′−1npBBnpBAk′−1−n∑m=0cτmk′,n,t+∑n=k′−ck′−1Ck′−1npBBnpBAk′−1−n∑m=0k′−1−nτmk′,n,t,and considering all possible values of k′, we can get the probability of iA (or iB) maintaining susceptible at time t:(14)ξASt=1−ε0∑k′=kmin'kmax'k′PAk′sA′k′,tkA′,ξBSt=1−ε0∑k′=kmin'kmax'k′PBk′sB′k′,tkB′,where kA′ (kB′) represents the average degree of iA (or iB). If iA (or iB) is in state A and does not transmit a message to j at time t−1, we can get(15)θAt=θAt−1−λξAAt−1,(16)θBt=θBt−1−λξBAt−1.ForξARt and ξBRt, two conditions need to be satisfied at time t to change them: (1) an adopted individual does not transmit a message through an edge with probability 1−λ; (2) this adopted individual recovers with probability γ. Thus(17)ξARt=ξARt−1+γ1−λξAAt−1,(18)ξBRt=ξBRt−1+γ1−λξBAt−1.In the initial stage,θA0=1, θB0=1 and ξAR0=0, ξBR0=0, for iA (or iB), combining equations (15) and (17) (or equations (16) and (18)), we can, respectively, get(19)ξARt=γ1−θAt1−λλ,(20)ξBRt=γ1−θBt1−λλ.Then combining equations (13), (22), and (26) (or (14), (23), and (20)), we can, respectively, get(21)ξAIt=θAt−1−ε0∑k′=kmin'kmax'k'PAk′sA′k′,tkA′−γ1−θAt1−λλ,ξBIt=θBt−1−ε0∑k′=kmin'kmax'k'PBk′sB′k′,tkB′−γ1−θBt1−λλ.Therefore, by combining equation (22) and (27) (or (23) and (29)), we can, respectively, get(22)θAt−θAt−1=−λθAt−1−1−ε0∑k′=kmin'kmax'k′PAk′sA′k′,t−1kA′+γ1−θAt1−λ,(23)θBt−θBt−1=−λθBt−1−1−ε0∑k′=kmin'kmax'k′PAk′sB′k′,t−1kB′+γ1−θBt1−λ.Whent⟶∞, the system tends to be stable; that is, limt⟶∞θAt−θAt−1=0, limt⟶∞θBt−θBt−1=0. At this time, there is no adopted individual in the system.We can get the final proportionS∞ of susceptible individuals by computing θA∞ and θB∞ based on the above process. And the final spreading size is R∞=1−S∞. To understand the effect of the influence strength of heterogeneous individuals and their contact preferences on social contagion dynamics, we need to analyze the change of root (θA∞, θB∞) of steady state: limt⟶∞θAt−θAt−1=0, limt⟶∞θBt−θBt−1=0, that is, the change of physically relevant root of(24)θA∞=yθA∞,θB∞=yθB∞,where(25)yθA∞=1−ε0∑k′=kmix'kmax'k′PAk′kA′sA′k′,t+γ1−θA∞1−λλ,yθB∞=1−ε0∑k′=kmin'kmax'k′PBk′kB′sB′k′,t+γ1−θB∞1−λλ.


### 3.2. Phase Transition Analysis

We assume(26)GAθA∞,θB∞=yθA∞−θA∞,GBθA∞,θB∞=yθB∞−θB∞.So,(27)GAθA∞,θB∞=0GBθA∞,θB∞=0.We can get the fixed points of system.We construct the network in which each type of individuals has the same degree value. When type-A individuals have the same degree values, we setkA to represent their degree values. Similarly, kB is used to represent the degree value when the type-B individuals’ degree values are the same. Thus, in this network kA = kB = 10, and other parameters are γ=1, ρB=0.19, and IB=T=5. We, respectively, analyze the phase transition in the case of p=0.9 and p=0.45 in this section.


#### 3.2.1. The Case ofp=0.9

Figure1 shows the change of roots (θA∞, θB∞) of equation (27) at p=0.9 when the proportion of seeds is vanishingly small. We analyze the stability of the roots by using Jacobi matrix J:(28)J=∂GAθA∞,θB∞∂θA∞∂GAθA∞,θB∞∂θB∞∂GBθA∞,θB∞∂θA∞∂GBθA∞,θB∞∂θB∞.Figure 1
The change of the roots (θA∞, θB∞) of equation (27) at p=0.9. The red and blue points are the roots of equation (27). The red points are physically relevant roots and the blue points are other roots. In the case of r0=1/5, the change of the roots of equation (27) at (a) λ = 0.1443, and the inset of (a) λ = 0.5, (b) λ = 0.745, and the inset of (c) λ = 0.78, (d) λ = 0.8, and the inset of (d) λ = 0.82. In the case of r0=3/5, the change of the roots of equation (27) at (e) λ = 0.1439, and the inset of (e) λ = 0.4, (f) λ = 0.7. (b) It shows the change of the roots of GBθA∞,θB∞ at θA∞=1; the inset of (b) shows the change of the roots of GAθA∞,θB∞ at θB∞=1, where the intersections of curves and horizontal line are the roots. Other parameters: ρB=0.19, IB=T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)Δ is the determinant of J. τ represents the trace of the matrix J. The main types of fixed points are saddle points, nodal points, and focus points. If Δ<0, the fixed point is saddle point. If Δ>0, the nodal point satisfies τ2−4Δ>0 and the focus point satisfies τ2−4Δ<0. However, if Δ>0, τ<0, the fixed point is stable [34]. The physically relevant roots which correspond to the steady state of the system are indicated by red points, and other roots are indicated by blue points.Figure1(a) shows that the physically relevant root (θA∞, θB∞) of (27) is the trivial root (1, 1), when p=0.9, r0=1/5, and λ is small. In this case, the information cannot be transmitted in the network. When λ is greater than the critical transmission probability λCII=0.1443 in the inset of Figure 1(a), the physically nontrivial root B occurs.In order to explore how to get the critical transmission probabilityλCII, we studied the variation of roots of GBθA∞,θB∞ at θA∞=1 in Figure 1(b), and the variation of roots of GAθA∞,θB∞ at θB∞=1 in the inset of Figure 1(b) to determine which type of individuals firstly plays a role and makes the physically nontrivial root occur. We find that the tangent point of curve GBθA∞,θB∞ at θB∞=1 occurs earlier than the tangent point of curve GAθA∞,θB∞ at θA∞=1 as λ increases. It reveals that type-B individuals play a role firstly and lead to the transmission of information. Thus, we can get the critical transmission probability by computing(29)∂GBθA∞,θB∞∂θB∞|θA∞=θB∞=1=0.With the increase ofλ, the physically relevant root changes continuously and a saddle-node C occurs in Figure 1(c). Then, the saddle point C bifurcates into two points which is shown in the inset of Figure 1(c). The point D which is bifurcated from C and nodal point B will collide at point E in Figure 1(d) when λ=λs, and then the point E vanishes (i.e., in the inset of Figure 1(d)) which makes the physically relevant root jumps to point A. We can get the critical transmission probability λs of discontinuous change in hybrid phase transition by observing the emergence of saddle-node bifurcation (the phenomenon that a saddle point and a nodal point collide and disappear is called a saddle-node bifurcation [34]) in theoretical analysis. Therefore, the spreading size of the system presents a hybrid phase transition with the transmission probability increases, when p=0.9; that is, the solution changes continuously at first and then discontinuously.In addition, we find another phase transition whenr0 is relatively large in Figures 2(e)–2(f). When λ is small, the physically relevant root is the trivial root (1, 1) in Figure 2(e) which reveals that the information cannot be transmitted in the system. When λ is greater than the critical transmission probability λCII=0.1439 as shown in the inset of Figure 1(e), the physically nontrivial root will be point B. As λ increases, the physically relevant root changes continuously which is shown in Figure 1(f). Therefore, the spreading size presents a continuous phase transition as the transmission probability increases, when p=0.9, r0=3/5.Figure 2
The change of the roots (θA∞, θB∞) of equation (27) at p=0.45. The red and blue points are the roots of equation (27). The red points are physically relevant roots and the blue points are other roots. In the case of r0=1/5, the change of the roots of equation (27) at (a) λ=0.7533, (b) λ=0.8, (d) λ=0.818, and the inset of (d) λ=0.9. (c) The change of the roots of GBθA∞,θB∞ at θA∞=1; the inset of (c) shows the change of the roots of GAθA∞,θB∞ at θB∞=1, where the intersections of curves and horizontal line are the roots. In the case of r0=3/5, the change of the roots of equation (27) at (e) λ=0.5, (f) λ=0.71, and the inset of (f) λ=0.83. Other parameters: ρB=0.19, IB=T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)


#### 3.2.2. The Case ofp=0.45

We also find a similar phenomenon in which the physically relevant root first changes continuously and then discontinuously whenp is small in Figures 2(a)–2(e)).Figure2(a) shows that physically relevant root (θA∞, θB∞) of equation (27) is the trivial root (1, 1) when p=0.45, r0=1/5, and λ is small. In this case, the information cannot be transmitted in the network. When λ is greater than the critical transmission probability λCII=0.7533 which can be got from equation (28) in Figure 2(b), the physically nontrivial root B occurs. Figure 2(c) reveals that type-B individuals play a role firstly and lead to the transmission of information.Asλ increases, the point C and point B will collide at point D in Figure 2(d) when λ=λs, and then the physically relevant root jumps to point A in the inset of Figure 2(d). We still can get the critical transmission probability λs of discontinuous change in hybrid phase transition by observing saddle-node bifurcation of the roots. Therefore, the spreading size of the system presents a hybrid phase transition with the transmission probability, when p=0.45, r0=1/5; that is, the physically relevant root changes continuously at first and then discontinuously.In addition, we find another phase transition whenr0 is relatively large in Figures 2(e) and 2(f). When λ is small, the physically relevant root is the trivial root B in Figure 2(e) which reveals that the information cannot be transmitted in the system. As λ increases, the point C moves close to point B and they collide at point D in Figure 2(f) when λ=λcI. Then the physically relevant root jumps to another outcome (i.e., point A in the inset of Figure 2(f)). We can get the critical transmission probability λcI in first-order phase transition by observing transcritical bifurcation (the two fixed points exchange stability after bifurcation and will not disappear [34]) of the roots. Therefore, the spreading size of the system presents a first-order phase transition with the transmission probability, when p=0.45, r0=3/5; that is, the physically relevant root changes discontinuously.


## 3.1. Spreading Process Analysis

Since the state change of susceptible individuals depends on the influence strength distribution and spreading states of neighbors, we use the edge-based compartmental theory to analyze the social contagion process with contact preference among heterogeneous individuals. Assuming that the influence strength of type-B individuals isIB=T, it means that they can persuade their neighbors directly. The ratio of influence strength between type-A and type-B individuals is expressed by r0=IA/IB (0<r0<1).Due to the existence of contact preference among heterogeneous individuals, the type distributions of neighbors of type-A and type-B individuals are different. Therefore, we will analyze the adoption processes of type-A and type-B individuals, respectively. For a random edge,θAt is defined as the probability that a type-A neighbor iA of individual j does not spread information to j until time t. θBt is defined as the probability that a type-B neighbor iB of individual j does not spread information to j until time t. Suppose that j has k neighbors, including n type-B individuals, and receives m messages until time t. The probability that all messages come from his/her k−n type-A neighbors is(1)ϕmk,n,t=Ck−nmθAtk−n−m1−θAtmθBtn.Ifj belongs to type-A, the probability that a randomly selected neighbor of j belongs to type-B (or type-A) is pAB (or pAA), and they satisfy pAB+pAA=1. If j belongs to type-B, the probability that a randomly selected neighbor of j belongs to type-B (or type-A) is pBB (or pBA), and they satisfy pBB+pBA=1. By counting the number of edges between the same type of individuals, we can get pAA and pBB in a given networks. Let sAk,n,t (or sBk,n,t) denote the probability that type-A (or type-B) individuals have n type-B neighbors and maintain susceptible at time t. Each individual can get at most c times of information from his/her type-A neighbors to maintain susceptible state. If k−n<c, then(2)sAk,n,t=CknpABnpAAk−n∑m=0k−nϕmk,n,t,sBk,n,t=CknpBBnpBAk−n∑m=0k−nϕmk,n,t.Ifk−n≥c, then(3)sAk,n,t=CknpABnpAAk−n∑m=0cϕmk,n,t,sBk,n,t=CknpBBnpBAk−n∑m=0cϕmk,n,t.Considering all possible values ofn, sAk,t (or sBk,t) represents the probability that a type-A (or type-B) individual with degree k is susceptible at time t. If k<c, then(4)sAk,t=∑n=0kCknpABnpAAk−n∑m=0k−nϕmk,n,t,sBk,t=∑n=0k−nCknpBBnpBAk−n∑m=0k−nϕmk,n,t.Ifk≥c, then(5)sAk,t=∑n=0k−cCknpABnpAAk−n∑m=0cϕmk,n,t+∑n=k−c+1kCknpABnpAAk−n∑m=0k−nϕmk,n,t,sBk,t=∑n=0k−cCknpBBnpBAk−n∑m=0cϕmk,n,t+∑n=k−c+1kCknpBBnpBAk−n∑m=0k−nϕmk,n,t.In the initial stage, a vanishingly small proportionε0 of individuals are randomly selected as seeds, the degree distributions of type-A (or type-B) individuals satisfy PAk (or PBk), and the susceptible proportion of type-A and type-B individuals at time t is, respectively,(6)SAt=1−ε0∑k=kminkmaxPAksAk,t,(7)SBt=1−ε0∑k=kminkmaxPBksBk,t,where (1 −ε0) represents the proportion of susceptible individuals in the initial state. Then the proportion of susceptible individuals in the network at time t is(8)St=ρASAt+ρBSBt.According to the above analysis, we can getSAt, SBt, and St by computing θAt and θBt. Here we will analyze how to compute θAt and θBt.According to the edge-based compartmental theory, since neighborsiA and iB of j may have possible three states: susceptible, adopted, and recovered state, we can express θAt and θBt as follows:(9)θAt=ξASt+ξAAt+ξARt,(10)θBt=ξBSt+ξBAt+ξBRt,where ξASt [ξAAt or ξARt] denotes the probability that a type-A neighbor iA is in S state [A state or R state] at time t and does not transmit information to j, and ξBSt [ξBAt or ξBRt] denotes the probability that a type-B neighbor is iB in S state [A state or R state] at time t and does not transmit information to j.We first analyzeξASt and ξBSt. When iA (or iB) is susceptible with k′ neighbors, he/she can receive messages from his/her k′−1 neighbors (as j which is susceptible and cannot transmit message). Assuming that, among the k′−1 neighbors of iA (or iB), there are n type-B individuals and k′−1−n type-A individuals, then iA (or iB) receives m messages at time t, and the probability that all the m messages come from his/her k′−1−n type-A neighbors is(11)τmk′,n,t=Ck′−1−nmθAtk′−1−n−m1−θAtmθBtn.Considering all possible values ofn, the probability of iA (or iB) with degree k′ maintaining susceptible at time t is as follows: if k′−1<c, then(12)sA′k′,t=∑n=0k′−1Ck′−1npABnpAAk′−1−n∑m=0k′−1−nτmk′,n,t,sB′k′,t=∑n=0k′−1Ck′−1npBBnpBAk′−1−n∑m=0k′−1−nτmk′,n,t.Ifk′−1≥c, then(13)sA′k′,t=∑n=0k′−1−cCk′−1npABnpAAk′−1−n∑m=0cτmk′,n,t+∑n=k′−ck′−1Ck′−1npABnpAAk′−1−n∑m=0k′−1−nτmk′,n,t,s′k′,t=∑n=0k′−1−cCk′−1npBBnpBAk′−1−n∑m=0cτmk′,n,t+∑n=k′−ck′−1Ck′−1npBBnpBAk′−1−n∑m=0k′−1−nτmk′,n,t,and considering all possible values of k′, we can get the probability of iA (or iB) maintaining susceptible at time t:(14)ξASt=1−ε0∑k′=kmin'kmax'k′PAk′sA′k′,tkA′,ξBSt=1−ε0∑k′=kmin'kmax'k′PBk′sB′k′,tkB′,where kA′ (kB′) represents the average degree of iA (or iB). If iA (or iB) is in state A and does not transmit a message to j at time t−1, we can get(15)θAt=θAt−1−λξAAt−1,(16)θBt=θBt−1−λξBAt−1.ForξARt and ξBRt, two conditions need to be satisfied at time t to change them: (1) an adopted individual does not transmit a message through an edge with probability 1−λ; (2) this adopted individual recovers with probability γ. Thus(17)ξARt=ξARt−1+γ1−λξAAt−1,(18)ξBRt=ξBRt−1+γ1−λξBAt−1.In the initial stage,θA0=1, θB0=1 and ξAR0=0, ξBR0=0, for iA (or iB), combining equations (15) and (17) (or equations (16) and (18)), we can, respectively, get(19)ξARt=γ1−θAt1−λλ,(20)ξBRt=γ1−θBt1−λλ.Then combining equations (13), (22), and (26) (or (14), (23), and (20)), we can, respectively, get(21)ξAIt=θAt−1−ε0∑k′=kmin'kmax'k'PAk′sA′k′,tkA′−γ1−θAt1−λλ,ξBIt=θBt−1−ε0∑k′=kmin'kmax'k'PBk′sB′k′,tkB′−γ1−θBt1−λλ.Therefore, by combining equation (22) and (27) (or (23) and (29)), we can, respectively, get(22)θAt−θAt−1=−λθAt−1−1−ε0∑k′=kmin'kmax'k′PAk′sA′k′,t−1kA′+γ1−θAt1−λ,(23)θBt−θBt−1=−λθBt−1−1−ε0∑k′=kmin'kmax'k′PAk′sB′k′,t−1kB′+γ1−θBt1−λ.Whent⟶∞, the system tends to be stable; that is, limt⟶∞θAt−θAt−1=0, limt⟶∞θBt−θBt−1=0. At this time, there is no adopted individual in the system.We can get the final proportionS∞ of susceptible individuals by computing θA∞ and θB∞ based on the above process. And the final spreading size is R∞=1−S∞. To understand the effect of the influence strength of heterogeneous individuals and their contact preferences on social contagion dynamics, we need to analyze the change of root (θA∞, θB∞) of steady state: limt⟶∞θAt−θAt−1=0, limt⟶∞θBt−θBt−1=0, that is, the change of physically relevant root of(24)θA∞=yθA∞,θB∞=yθB∞,where(25)yθA∞=1−ε0∑k′=kmix'kmax'k′PAk′kA′sA′k′,t+γ1−θA∞1−λλ,yθB∞=1−ε0∑k′=kmin'kmax'k′PBk′kB′sB′k′,t+γ1−θB∞1−λλ.


## 3.2. Phase Transition Analysis

We assume(26)GAθA∞,θB∞=yθA∞−θA∞,GBθA∞,θB∞=yθB∞−θB∞.So,(27)GAθA∞,θB∞=0GBθA∞,θB∞=0.We can get the fixed points of system.We construct the network in which each type of individuals has the same degree value. When type-A individuals have the same degree values, we setkA to represent their degree values. Similarly, kB is used to represent the degree value when the type-B individuals’ degree values are the same. Thus, in this network kA = kB = 10, and other parameters are γ=1, ρB=0.19, and IB=T=5. We, respectively, analyze the phase transition in the case of p=0.9 and p=0.45 in this section.


### 3.2.1. The Case ofp=0.9

Figure1 shows the change of roots (θA∞, θB∞) of equation (27) at p=0.9 when the proportion of seeds is vanishingly small. We analyze the stability of the roots by using Jacobi matrix J:(28)J=∂GAθA∞,θB∞∂θA∞∂GAθA∞,θB∞∂θB∞∂GBθA∞,θB∞∂θA∞∂GBθA∞,θB∞∂θB∞.Figure 1
The change of the roots (θA∞, θB∞) of equation (27) at p=0.9. The red and blue points are the roots of equation (27). The red points are physically relevant roots and the blue points are other roots. In the case of r0=1/5, the change of the roots of equation (27) at (a) λ = 0.1443, and the inset of (a) λ = 0.5, (b) λ = 0.745, and the inset of (c) λ = 0.78, (d) λ = 0.8, and the inset of (d) λ = 0.82. In the case of r0=3/5, the change of the roots of equation (27) at (e) λ = 0.1439, and the inset of (e) λ = 0.4, (f) λ = 0.7. (b) It shows the change of the roots of GBθA∞,θB∞ at θA∞=1; the inset of (b) shows the change of the roots of GAθA∞,θB∞ at θB∞=1, where the intersections of curves and horizontal line are the roots. Other parameters: ρB=0.19, IB=T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)Δ is the determinant of J. τ represents the trace of the matrix J. The main types of fixed points are saddle points, nodal points, and focus points. If Δ<0, the fixed point is saddle point. If Δ>0, the nodal point satisfies τ2−4Δ>0 and the focus point satisfies τ2−4Δ<0. However, if Δ>0, τ<0, the fixed point is stable [34]. The physically relevant roots which correspond to the steady state of the system are indicated by red points, and other roots are indicated by blue points.Figure1(a) shows that the physically relevant root (θA∞, θB∞) of (27) is the trivial root (1, 1), when p=0.9, r0=1/5, and λ is small. In this case, the information cannot be transmitted in the network. When λ is greater than the critical transmission probability λCII=0.1443 in the inset of Figure 1(a), the physically nontrivial root B occurs.In order to explore how to get the critical transmission probabilityλCII, we studied the variation of roots of GBθA∞,θB∞ at θA∞=1 in Figure 1(b), and the variation of roots of GAθA∞,θB∞ at θB∞=1 in the inset of Figure 1(b) to determine which type of individuals firstly plays a role and makes the physically nontrivial root occur. We find that the tangent point of curve GBθA∞,θB∞ at θB∞=1 occurs earlier than the tangent point of curve GAθA∞,θB∞ at θA∞=1 as λ increases. It reveals that type-B individuals play a role firstly and lead to the transmission of information. Thus, we can get the critical transmission probability by computing(29)∂GBθA∞,θB∞∂θB∞|θA∞=θB∞=1=0.With the increase ofλ, the physically relevant root changes continuously and a saddle-node C occurs in Figure 1(c). Then, the saddle point C bifurcates into two points which is shown in the inset of Figure 1(c). The point D which is bifurcated from C and nodal point B will collide at point E in Figure 1(d) when λ=λs, and then the point E vanishes (i.e., in the inset of Figure 1(d)) which makes the physically relevant root jumps to point A. We can get the critical transmission probability λs of discontinuous change in hybrid phase transition by observing the emergence of saddle-node bifurcation (the phenomenon that a saddle point and a nodal point collide and disappear is called a saddle-node bifurcation [34]) in theoretical analysis. Therefore, the spreading size of the system presents a hybrid phase transition with the transmission probability increases, when p=0.9; that is, the solution changes continuously at first and then discontinuously.In addition, we find another phase transition whenr0 is relatively large in Figures 2(e)–2(f). When λ is small, the physically relevant root is the trivial root (1, 1) in Figure 2(e) which reveals that the information cannot be transmitted in the system. When λ is greater than the critical transmission probability λCII=0.1439 as shown in the inset of Figure 1(e), the physically nontrivial root will be point B. As λ increases, the physically relevant root changes continuously which is shown in Figure 1(f). Therefore, the spreading size presents a continuous phase transition as the transmission probability increases, when p=0.9, r0=3/5.Figure 2
The change of the roots (θA∞, θB∞) of equation (27) at p=0.45. The red and blue points are the roots of equation (27). The red points are physically relevant roots and the blue points are other roots. In the case of r0=1/5, the change of the roots of equation (27) at (a) λ=0.7533, (b) λ=0.8, (d) λ=0.818, and the inset of (d) λ=0.9. (c) The change of the roots of GBθA∞,θB∞ at θA∞=1; the inset of (c) shows the change of the roots of GAθA∞,θB∞ at θB∞=1, where the intersections of curves and horizontal line are the roots. In the case of r0=3/5, the change of the roots of equation (27) at (e) λ=0.5, (f) λ=0.71, and the inset of (f) λ=0.83. Other parameters: ρB=0.19, IB=T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)


### 3.2.2. The Case ofp=0.45

We also find a similar phenomenon in which the physically relevant root first changes continuously and then discontinuously whenp is small in Figures 2(a)–2(e)).Figure2(a) shows that physically relevant root (θA∞, θB∞) of equation (27) is the trivial root (1, 1) when p=0.45, r0=1/5, and λ is small. In this case, the information cannot be transmitted in the network. When λ is greater than the critical transmission probability λCII=0.7533 which can be got from equation (28) in Figure 2(b), the physically nontrivial root B occurs. Figure 2(c) reveals that type-B individuals play a role firstly and lead to the transmission of information.Asλ increases, the point C and point B will collide at point D in Figure 2(d) when λ=λs, and then the physically relevant root jumps to point A in the inset of Figure 2(d). We still can get the critical transmission probability λs of discontinuous change in hybrid phase transition by observing saddle-node bifurcation of the roots. Therefore, the spreading size of the system presents a hybrid phase transition with the transmission probability, when p=0.45, r0=1/5; that is, the physically relevant root changes continuously at first and then discontinuously.In addition, we find another phase transition whenr0 is relatively large in Figures 2(e) and 2(f). When λ is small, the physically relevant root is the trivial root B in Figure 2(e) which reveals that the information cannot be transmitted in the system. As λ increases, the point C moves close to point B and they collide at point D in Figure 2(f) when λ=λcI. Then the physically relevant root jumps to another outcome (i.e., point A in the inset of Figure 2(f)). We can get the critical transmission probability λcI in first-order phase transition by observing transcritical bifurcation (the two fixed points exchange stability after bifurcation and will not disappear [34]) of the roots. Therefore, the spreading size of the system presents a first-order phase transition with the transmission probability, when p=0.45, r0=3/5; that is, the physically relevant root changes discontinuously.


## 3.2.1. The Case ofp=0.9

Figure1 shows the change of roots (θA∞, θB∞) of equation (27) at p=0.9 when the proportion of seeds is vanishingly small. We analyze the stability of the roots by using Jacobi matrix J:(28)J=∂GAθA∞,θB∞∂θA∞∂GAθA∞,θB∞∂θB∞∂GBθA∞,θB∞∂θA∞∂GBθA∞,θB∞∂θB∞.Figure 1
The change of the roots (θA∞, θB∞) of equation (27) at p=0.9. The red and blue points are the roots of equation (27). The red points are physically relevant roots and the blue points are other roots. In the case of r0=1/5, the change of the roots of equation (27) at (a) λ = 0.1443, and the inset of (a) λ = 0.5, (b) λ = 0.745, and the inset of (c) λ = 0.78, (d) λ = 0.8, and the inset of (d) λ = 0.82. In the case of r0=3/5, the change of the roots of equation (27) at (e) λ = 0.1439, and the inset of (e) λ = 0.4, (f) λ = 0.7. (b) It shows the change of the roots of GBθA∞,θB∞ at θA∞=1; the inset of (b) shows the change of the roots of GAθA∞,θB∞ at θB∞=1, where the intersections of curves and horizontal line are the roots. Other parameters: ρB=0.19, IB=T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)Δ is the determinant of J. τ represents the trace of the matrix J. The main types of fixed points are saddle points, nodal points, and focus points. If Δ<0, the fixed point is saddle point. If Δ>0, the nodal point satisfies τ2−4Δ>0 and the focus point satisfies τ2−4Δ<0. However, if Δ>0, τ<0, the fixed point is stable [34]. The physically relevant roots which correspond to the steady state of the system are indicated by red points, and other roots are indicated by blue points.Figure1(a) shows that the physically relevant root (θA∞, θB∞) of (27) is the trivial root (1, 1), when p=0.9, r0=1/5, and λ is small. In this case, the information cannot be transmitted in the network. When λ is greater than the critical transmission probability λCII=0.1443 in the inset of Figure 1(a), the physically nontrivial root B occurs.In order to explore how to get the critical transmission probabilityλCII, we studied the variation of roots of GBθA∞,θB∞ at θA∞=1 in Figure 1(b), and the variation of roots of GAθA∞,θB∞ at θB∞=1 in the inset of Figure 1(b) to determine which type of individuals firstly plays a role and makes the physically nontrivial root occur. We find that the tangent point of curve GBθA∞,θB∞ at θB∞=1 occurs earlier than the tangent point of curve GAθA∞,θB∞ at θA∞=1 as λ increases. It reveals that type-B individuals play a role firstly and lead to the transmission of information. Thus, we can get the critical transmission probability by computing(29)∂GBθA∞,θB∞∂θB∞|θA∞=θB∞=1=0.With the increase ofλ, the physically relevant root changes continuously and a saddle-node C occurs in Figure 1(c). Then, the saddle point C bifurcates into two points which is shown in the inset of Figure 1(c). The point D which is bifurcated from C and nodal point B will collide at point E in Figure 1(d) when λ=λs, and then the point E vanishes (i.e., in the inset of Figure 1(d)) which makes the physically relevant root jumps to point A. We can get the critical transmission probability λs of discontinuous change in hybrid phase transition by observing the emergence of saddle-node bifurcation (the phenomenon that a saddle point and a nodal point collide and disappear is called a saddle-node bifurcation [34]) in theoretical analysis. Therefore, the spreading size of the system presents a hybrid phase transition with the transmission probability increases, when p=0.9; that is, the solution changes continuously at first and then discontinuously.In addition, we find another phase transition whenr0 is relatively large in Figures 2(e)–2(f). When λ is small, the physically relevant root is the trivial root (1, 1) in Figure 2(e) which reveals that the information cannot be transmitted in the system. When λ is greater than the critical transmission probability λCII=0.1439 as shown in the inset of Figure 1(e), the physically nontrivial root will be point B. As λ increases, the physically relevant root changes continuously which is shown in Figure 1(f). Therefore, the spreading size presents a continuous phase transition as the transmission probability increases, when p=0.9, r0=3/5.Figure 2
The change of the roots (θA∞, θB∞) of equation (27) at p=0.45. The red and blue points are the roots of equation (27). The red points are physically relevant roots and the blue points are other roots. In the case of r0=1/5, the change of the roots of equation (27) at (a) λ=0.7533, (b) λ=0.8, (d) λ=0.818, and the inset of (d) λ=0.9. (c) The change of the roots of GBθA∞,θB∞ at θA∞=1; the inset of (c) shows the change of the roots of GAθA∞,θB∞ at θB∞=1, where the intersections of curves and horizontal line are the roots. In the case of r0=3/5, the change of the roots of equation (27) at (e) λ=0.5, (f) λ=0.71, and the inset of (f) λ=0.83. Other parameters: ρB=0.19, IB=T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)


## 3.2.2. The Case ofp=0.45

We also find a similar phenomenon in which the physically relevant root first changes continuously and then discontinuously whenp is small in Figures 2(a)–2(e)).Figure2(a) shows that physically relevant root (θA∞, θB∞) of equation (27) is the trivial root (1, 1) when p=0.45, r0=1/5, and λ is small. In this case, the information cannot be transmitted in the network. When λ is greater than the critical transmission probability λCII=0.7533 which can be got from equation (28) in Figure 2(b), the physically nontrivial root B occurs. Figure 2(c) reveals that type-B individuals play a role firstly and lead to the transmission of information.Asλ increases, the point C and point B will collide at point D in Figure 2(d) when λ=λs, and then the physically relevant root jumps to point A in the inset of Figure 2(d). We still can get the critical transmission probability λs of discontinuous change in hybrid phase transition by observing saddle-node bifurcation of the roots. Therefore, the spreading size of the system presents a hybrid phase transition with the transmission probability, when p=0.45, r0=1/5; that is, the physically relevant root changes continuously at first and then discontinuously.In addition, we find another phase transition whenr0 is relatively large in Figures 2(e) and 2(f). When λ is small, the physically relevant root is the trivial root B in Figure 2(e) which reveals that the information cannot be transmitted in the system. As λ increases, the point C moves close to point B and they collide at point D in Figure 2(f) when λ=λcI. Then the physically relevant root jumps to another outcome (i.e., point A in the inset of Figure 2(f)). We can get the critical transmission probability λcI in first-order phase transition by observing transcritical bifurcation (the two fixed points exchange stability after bifurcation and will not disappear [34]) of the roots. Therefore, the spreading size of the system presents a first-order phase transition with the transmission probability, when p=0.45, r0=3/5; that is, the physically relevant root changes discontinuously.


## 4. Numerical Verification

To explore the influence of contact preference on the spreading size and verify the theoretical predictions, we have extensively simulated the social contagion processes in networks with average degreekA = kB = 10. Each simulation result was averaged 1000 times. We first verify the above theoretical analysis results and then study what kind of phase transition will happen in networks with different contact preferences. Finally, we explore the optimal contact preference situation which maximizes the spreading size.


### 4.1. Role of Heterogeneous Individuals with Contact Preference

In order to verify the accuracy of the theoretical analysis of Section3, we give the simulation and theoretical results of the spreading size under different distributions of heterogeneous individuals and influence strength ratio in Figure 3. The results show that theoretical and simulation results are in good agreement. In addition, we also explore how heterogeneous individuals affect the phase transition in the insets of Figure 3, where ρtotalA∞ represents the adoption size only affected by type-A individuals, ρtotalB∞ represents the adoption size only affected by type-B individuals, and ρtotalmix∞ represents the adoption size affected by the two types of individuals together.Figure 3
The change of spreading size (R) (∞) with the increase ofλ. The red circle represents the proportion (R) (∞) of the adoption size of individuals, the blue square represents the proportion RA∞ of the adoption size of type-A individuals, and the green triangle represents the proportion RB∞ of the adoption size of type-B individuals. (a–d) when r0=1/5, p=0.45, 0.7,0.9 and 0.975, the adoption size changes with the increase of λ. (e–f) when r0=3/5, p=0.45,0.7, the adoption size changes with the increase of λ. Symbols represent simulation results and curves represent theoretical numerical results, which are calculated by equations (10), (11), (22), and (23). The insets show the changes of the proportion of adopted individuals affected only by type-A individuals (blue point) and only by type-B individuals (green point) and the proportion of adopted individuals affected by the two types of individuals together (red point). The vertical black lines represent λ=λcII, the vertical orange lines represent λ=λs, and the vertical purple lines represent λ=λcI. Other parameters: N=10000, ρB = 0.19, IB = 5, T=5, kA =  kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)We first studied the situation that the influence strength of ordinary individuals is low (i.e., Figures3(a)–3(d)). In this case, there is an obvious social reinforcement phenomenon in the system; that is, an individual needs multiple reinforcements from ordinary individuals to be persuaded (that is, the complex contagion).Whenp is low (e.g., p=0.45 in Figure 3(a)), individuals are more likely to interact with other types of neighbors and the hybrid phase transition occurs. Firstly, ρtotalB∞ grows continuously because of type-B individuals affecting other individuals directly (that is, the simple contagion) at the earliest. With the increase of λ, more and more type-A individuals are affected by their type-B neighbors and become adopted; thus they play a more important role and the complex contagion dominates (i.e., type-A individuals playing roles), which make the spreading size jump to another value.Whenp is relatively high (e.g., p=0.7 in Figure 3(b)), ordinary individuals tend to interact with the same type of neighbors but still have a part of connections with individuals of high influence strength. In this case, ρtotalB∞ grows continuously and is larger than ρtotalA∞ and ρtotalmix∞, which means the contagion relies mainly on type-B individuals. At this time, simple contagion dominates as is shown in the inset of Figure 3(b), and the spreading size varies continuously with the increase of λ.Whenp=0.9 (e.g., Figure 3(c)), individuals are more likely to interact with the same type of neighbors, and there are only a small number of connections among different types of individuals. In this case, hybrid phase transition occurs. Firstly, type-B individuals play a role and affect the same type of individuals which leads to the continuous grow of R (∞). When λ is large enough, ρtotalA∞ jumps to another value and is larger than ρtotalB∞, which means that type-A individuals are affected by type-B individuals and then can play a sufficient role in affecting the individuals within the type-A cluster. At this time, complex contagion dominates as shown in the inset of Figure 3(c) and leads to the discontinuous change of spreading size.However, when the two types of individuals are basically disconnected (i.e.,p=0.975 in Figure 3(d)), it is difficult for ordinary individuals to play a role because of their low influence strength. Therefore, the contagion mainly occurs in the type-B cluster and the spreading size grows continuously with the increase of λ.When the influence strength of ordinary individuals is relatively high (i.e., Figures3(e) and 3(f)), social reinforcement phenomenon is weakened. When ordinary individuals tend to interact with individuals of high influence (e.g., p=0.45 in Figure 3(e)), type-A individuals can be affected by type-B individuals easily. At this time, type-A individuals can play a sufficient role and lead to the occurrence of first-order phase transition of the system. When individuals tend to interact with the same type of individuals (e.g., p=0.7 in Figure 3(f)), type-B individuals tend to play a role firstly and then drive type-A individuals together to spread the information. At this time, the system shows continuous phase transition.


### 4.2. Effect of Contact Preference on Phase Transition

In the above section, we find that different distributions of contact probability of homogeneous individuals and influence strength lead to various phase transitions, so we study the phase transitions in the parameter planep,λ in Figure 4.Figure 4
The change of phase transition. Dependence ofR∞ on (p) and λ at r0=1/5 in (a) and (b) and at r0=3/5 in (c) and (d). The color-coded values of final spreading size R∞ in the parameter plane p,λ of (a) and (c) are from simulation results, (b) and (d) are from theoretical predictions. The theoretical predictions are the solutions of equations (12) and (22)–(24). The different phase transition regions are separated by the vertical lines which represent the critical contact probability of homogeneous individuals. Other parameters: N=10000, ρB = 0.19, IB = 5, T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)For the influence strength ratior0=1/5 (i.e., Figures 4(a) and 4(b)), the parameter plane is divided into five regions. In the region I of p≤pc∗≈0.38 (the white lines), information cannot be spread for any value of λ. In the region II of 0.38<p≤0.58 (the purple lines), the physically relevant root of equation (27) first changes continuously and then jumps to another certain value. This region is hybrid phase transition. In the region III of 0.58<p≤0.87 (the green lines), the physically relevant root of equation (27) changes continuously with the increase of λ. This region is continuous phase transition. In the region IV of 0.87<p≤0.95 (the yellow lines), the physically relevant root of equation (27) first changes continuously and then jumps to a certain value. This region is hybrid phase transition. In the region V of 0.95<p≤1, the physically relevant root of equation (27) changes continuously with the increase of λ but cannot reach global transmission. This region is continuous phase transition.For the influence strength ratior0=3/5 (i.e., Figures 4(c) and 4(d)), the parameter plane is divided into three regions. In the region I of p≤pc∗≈0.41 (the white lines), the information cannot be spread globally for any value of λ. In the region II of 0.41<p≤0.59 (the purple lines), the physically relevant root (θA∞, θB∞) of equation (27) is (1, 1) and then jumps to another certain value. This region is first-order phase transition. In the region III of 0.59<p≤1, the physically relevant root of equation (27) changes continuously with the increase of λ. This region is continuous phase transition.Here we can find that the greater the contact probability is, the smaller the transmission threshold is. However, the greater value of contact probability does not mean the larger spreading size. Thus, we will explore the optimal contact probability which allows the spreading size to reach the maximum value.


### 4.3. Exploring the Optimal Contact Preference

Figure5 shows how the spreading size changes versus contact probability of homogeneous individuals under different influence strength ratio r0, proportion ρB of type-B individuals, and degree distributions. We find that there is an optimal contact probability value or range which maximizes the spreading size, and the optimal contact probability value or lower bound of the optimal range decreases with the increase of transmission probability.Figure 5
The change of spreading sizeR∞ versus contact probability p of homogeneous individuals. (a–d) The individuals’ degree distribution satisfies kA=kB=10, where (a) r0=1/5, ρB=0.19. (b) r0=3/5, ρB=0.19. (c) r0=1/5, ρB=0.3. (d) r0=3/5, ρB=0.3. (e) Case of kA=kB=10, where r0=1/5, ρB=0.19. (f) Case of kA=8, kB=19, where r0=1/5, ρB=0.19. Symbols represent simulation results and lines represent theoretical prediction results. Other parameters: N=10000, IB = 5, T=5, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)We first discussed the network in which each of the individuals has the same degree valueskA=kB=10. When the influence strength ratio is small (i.e., r0=1/5 in Figures 5(a) and 5(b)), we find that there is an optimal contact probability p when λ is small (e.g., λ=0.2,0.4,0.6). According to the stimulation results, the optimal contact probability, respectively, is near p=0.95,0.75,0.65 at λ=0.2,0.4,0.6 when ρB=0.19. When ρB=0.3, the optimal contact probability value, respectively, is near p=0.9,0.7,0.55 at λ=0.2,0.4,0.6. We can get that the larger λ leads to the smaller value of optimal p. In the case of large λ (e.g., λ=0.8), there is an optimal range of contact probability, which makes the spreading size reach global propagation.When the influence strength ratio is relatively large (i.e.,r0=3/5 in Figures 5(c) and 5(d)), we also find similar results. There is an optimal contact probability p when λ is small (e.g., λ=0.2). In the case of large λ (e.g., λ=0.4,0.6,0.8), there is an optimal range of contact probability, which makes the spreading size reach the maximum value. With the increase of λ, the lower bound of the range decreases, and the range length increases.We also discussed another degree distribution of two types of individuals and found similar results. We can get that there is an optimal contact preference value or range that maximizes the spreading size, and the optimal value decreases with the increase of transmission probability. WhenPAk and PBk satisfy Poisson distribution and the average degree value is kA=kB=10 (i.e., Figure 5(e)), the optimal contact probability, respectively, is near p=0.95,0.75,0.65,0.4 at λ=0.2,0.4,0.6,0.8. When PAk and PBk satisfy Poisson distribution and the average degree value, respectively, is kA=8, kB=19 (i.e., Figure 5(f)), the optimal contact probability, respectively, is near p=0.75,0.3,0.225,0.2 at λ=0.2,0.4,0.6, 0.8.


## 4.1. Role of Heterogeneous Individuals with Contact Preference

In order to verify the accuracy of the theoretical analysis of Section3, we give the simulation and theoretical results of the spreading size under different distributions of heterogeneous individuals and influence strength ratio in Figure 3. The results show that theoretical and simulation results are in good agreement. In addition, we also explore how heterogeneous individuals affect the phase transition in the insets of Figure 3, where ρtotalA∞ represents the adoption size only affected by type-A individuals, ρtotalB∞ represents the adoption size only affected by type-B individuals, and ρtotalmix∞ represents the adoption size affected by the two types of individuals together.Figure 3
The change of spreading size (R) (∞) with the increase ofλ. The red circle represents the proportion (R) (∞) of the adoption size of individuals, the blue square represents the proportion RA∞ of the adoption size of type-A individuals, and the green triangle represents the proportion RB∞ of the adoption size of type-B individuals. (a–d) when r0=1/5, p=0.45, 0.7,0.9 and 0.975, the adoption size changes with the increase of λ. (e–f) when r0=3/5, p=0.45,0.7, the adoption size changes with the increase of λ. Symbols represent simulation results and curves represent theoretical numerical results, which are calculated by equations (10), (11), (22), and (23). The insets show the changes of the proportion of adopted individuals affected only by type-A individuals (blue point) and only by type-B individuals (green point) and the proportion of adopted individuals affected by the two types of individuals together (red point). The vertical black lines represent λ=λcII, the vertical orange lines represent λ=λs, and the vertical purple lines represent λ=λcI. Other parameters: N=10000, ρB = 0.19, IB = 5, T=5, kA =  kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)We first studied the situation that the influence strength of ordinary individuals is low (i.e., Figures3(a)–3(d)). In this case, there is an obvious social reinforcement phenomenon in the system; that is, an individual needs multiple reinforcements from ordinary individuals to be persuaded (that is, the complex contagion).Whenp is low (e.g., p=0.45 in Figure 3(a)), individuals are more likely to interact with other types of neighbors and the hybrid phase transition occurs. Firstly, ρtotalB∞ grows continuously because of type-B individuals affecting other individuals directly (that is, the simple contagion) at the earliest. With the increase of λ, more and more type-A individuals are affected by their type-B neighbors and become adopted; thus they play a more important role and the complex contagion dominates (i.e., type-A individuals playing roles), which make the spreading size jump to another value.Whenp is relatively high (e.g., p=0.7 in Figure 3(b)), ordinary individuals tend to interact with the same type of neighbors but still have a part of connections with individuals of high influence strength. In this case, ρtotalB∞ grows continuously and is larger than ρtotalA∞ and ρtotalmix∞, which means the contagion relies mainly on type-B individuals. At this time, simple contagion dominates as is shown in the inset of Figure 3(b), and the spreading size varies continuously with the increase of λ.Whenp=0.9 (e.g., Figure 3(c)), individuals are more likely to interact with the same type of neighbors, and there are only a small number of connections among different types of individuals. In this case, hybrid phase transition occurs. Firstly, type-B individuals play a role and affect the same type of individuals which leads to the continuous grow of R (∞). When λ is large enough, ρtotalA∞ jumps to another value and is larger than ρtotalB∞, which means that type-A individuals are affected by type-B individuals and then can play a sufficient role in affecting the individuals within the type-A cluster. At this time, complex contagion dominates as shown in the inset of Figure 3(c) and leads to the discontinuous change of spreading size.However, when the two types of individuals are basically disconnected (i.e.,p=0.975 in Figure 3(d)), it is difficult for ordinary individuals to play a role because of their low influence strength. Therefore, the contagion mainly occurs in the type-B cluster and the spreading size grows continuously with the increase of λ.When the influence strength of ordinary individuals is relatively high (i.e., Figures3(e) and 3(f)), social reinforcement phenomenon is weakened. When ordinary individuals tend to interact with individuals of high influence (e.g., p=0.45 in Figure 3(e)), type-A individuals can be affected by type-B individuals easily. At this time, type-A individuals can play a sufficient role and lead to the occurrence of first-order phase transition of the system. When individuals tend to interact with the same type of individuals (e.g., p=0.7 in Figure 3(f)), type-B individuals tend to play a role firstly and then drive type-A individuals together to spread the information. At this time, the system shows continuous phase transition.


## 4.2. Effect of Contact Preference on Phase Transition

In the above section, we find that different distributions of contact probability of homogeneous individuals and influence strength lead to various phase transitions, so we study the phase transitions in the parameter planep,λ in Figure 4.Figure 4
The change of phase transition. Dependence ofR∞ on (p) and λ at r0=1/5 in (a) and (b) and at r0=3/5 in (c) and (d). The color-coded values of final spreading size R∞ in the parameter plane p,λ of (a) and (c) are from simulation results, (b) and (d) are from theoretical predictions. The theoretical predictions are the solutions of equations (12) and (22)–(24). The different phase transition regions are separated by the vertical lines which represent the critical contact probability of homogeneous individuals. Other parameters: N=10000, ρB = 0.19, IB = 5, T=5, kA = kB = 10, γ=1, ε0⟶0.
(a)(b)(c)(d)For the influence strength ratior0=1/5 (i.e., Figures 4(a) and 4(b)), the parameter plane is divided into five regions. In the region I of p≤pc∗≈0.38 (the white lines), information cannot be spread for any value of λ. In the region II of 0.38<p≤0.58 (the purple lines), the physically relevant root of equation (27) first changes continuously and then jumps to another certain value. This region is hybrid phase transition. In the region III of 0.58<p≤0.87 (the green lines), the physically relevant root of equation (27) changes continuously with the increase of λ. This region is continuous phase transition. In the region IV of 0.87<p≤0.95 (the yellow lines), the physically relevant root of equation (27) first changes continuously and then jumps to a certain value. This region is hybrid phase transition. In the region V of 0.95<p≤1, the physically relevant root of equation (27) changes continuously with the increase of λ but cannot reach global transmission. This region is continuous phase transition.For the influence strength ratior0=3/5 (i.e., Figures 4(c) and 4(d)), the parameter plane is divided into three regions. In the region I of p≤pc∗≈0.41 (the white lines), the information cannot be spread globally for any value of λ. In the region II of 0.41<p≤0.59 (the purple lines), the physically relevant root (θA∞, θB∞) of equation (27) is (1, 1) and then jumps to another certain value. This region is first-order phase transition. In the region III of 0.59<p≤1, the physically relevant root of equation (27) changes continuously with the increase of λ. This region is continuous phase transition.Here we can find that the greater the contact probability is, the smaller the transmission threshold is. However, the greater value of contact probability does not mean the larger spreading size. Thus, we will explore the optimal contact probability which allows the spreading size to reach the maximum value.


## 4.3. Exploring the Optimal Contact Preference

Figure5 shows how the spreading size changes versus contact probability of homogeneous individuals under different influence strength ratio r0, proportion ρB of type-B individuals, and degree distributions. We find that there is an optimal contact probability value or range which maximizes the spreading size, and the optimal contact probability value or lower bound of the optimal range decreases with the increase of transmission probability.Figure 5
The change of spreading sizeR∞ versus contact probability p of homogeneous individuals. (a–d) The individuals’ degree distribution satisfies kA=kB=10, where (a) r0=1/5, ρB=0.19. (b) r0=3/5, ρB=0.19. (c) r0=1/5, ρB=0.3. (d) r0=3/5, ρB=0.3. (e) Case of kA=kB=10, where r0=1/5, ρB=0.19. (f) Case of kA=8, kB=19, where r0=1/5, ρB=0.19. Symbols represent simulation results and lines represent theoretical prediction results. Other parameters: N=10000, IB = 5, T=5, γ=1, ε0⟶0.
(a)(b)(c)(d)(e)(f)We first discussed the network in which each of the individuals has the same degree valueskA=kB=10. When the influence strength ratio is small (i.e., r0=1/5 in Figures 5(a) and 5(b)), we find that there is an optimal contact probability p when λ is small (e.g., λ=0.2,0.4,0.6). According to the stimulation results, the optimal contact probability, respectively, is near p=0.95,0.75,0.65 at λ=0.2,0.4,0.6 when ρB=0.19. When ρB=0.3, the optimal contact probability value, respectively, is near p=0.9,0.7,0.55 at λ=0.2,0.4,0.6. We can get that the larger λ leads to the smaller value of optimal p. In the case of large λ (e.g., λ=0.8), there is an optimal range of contact probability, which makes the spreading size reach global propagation.When the influence strength ratio is relatively large (i.e.,r0=3/5 in Figures 5(c) and 5(d)), we also find similar results. There is an optimal contact probability p when λ is small (e.g., λ=0.2). In the case of large λ (e.g., λ=0.4,0.6,0.8), there is an optimal range of contact probability, which makes the spreading size reach the maximum value. With the increase of λ, the lower bound of the range decreases, and the range length increases.We also discussed another degree distribution of two types of individuals and found similar results. We can get that there is an optimal contact preference value or range that maximizes the spreading size, and the optimal value decreases with the increase of transmission probability. WhenPAk and PBk satisfy Poisson distribution and the average degree value is kA=kB=10 (i.e., Figure 5(e)), the optimal contact probability, respectively, is near p=0.95,0.75,0.65,0.4 at λ=0.2,0.4,0.6,0.8. When PAk and PBk satisfy Poisson distribution and the average degree value, respectively, is kA=8, kB=19 (i.e., Figure 5(f)), the optimal contact probability, respectively, is near p=0.75,0.3,0.225,0.2 at λ=0.2,0.4,0.6, 0.8.


## 5. Conclusion

Individuals' spreading behavior in real social networks is not randomly distributed but has certain tendency. In order to explore the influence of contact preference among heterogeneous individuals on spreading size and phase transition, we construct a social contagion model. Through simulating and theoretical analysis, we study how individuals with heterogeneous influence strength and contact preference affect the spreading process. We can get that the competition between simple contagion and complex contagion causes the crossover phase transitions. It changes from the hybrid to continuous phase transition, then to the hybrid, and eventually to continuous phase transition when the influence strength of ordinary individuals is low. When the influence strength of ordinary individuals is relatively high, however, it changes from being first-order to continuous phase transition. Then, we discuss the phase transition in parameter planep,λ and find that the larger the contact probability is, the smaller the transmission threshold is. However, it is not that greater value of the contact probability leads to the larger value of spreading size. There is an optimal value or range of contact probability of homogeneous individuals which maximizes the spreading size, and the value or lower bound of the range decreases with the increase of transmission probability.Our results provide a theoretical reference for the promotion of new knowledge/products or the control of information. For example, in the promotion of new products, our results can help formulate promotion strategies. When there is a small number of authoritative individuals and they have limited contact capacity, our results show that if the authoritative individuals mainly contact with ordinary individuals, the product is difficult to be further spread by ordinary individuals, as ordinary individuals have little influence. Similarly, it is not conducive to the promotion of the product when the two types of individuals spread separately in their own communities. Therefore, authoritative individuals are required to contact approximately other ordinary individuals, and the contact probability mainly depends on the transmission probability of the product and network structure. Our results provide a theoretical reference for the promotion of new knowledge/products or the control of information.In the future research work, how to quantify contact preferences among heterogeneous individuals in different types of information transmission through real data, so as to better guide the model construction, should be considered.



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*Source: 4471361-2022-01-29.xml*