Dynamics of social contagions with heterogeneous adoption thresholds: crossover phenomena in phase transition

Individual u is set to be in the cavity state, which means that he can receive behavioral information from neighbors but not transmit behavioral information to his neighbors [44]. And $\theta (t)$ is defined to be the probability that individual v has not transmitted the behavioral information to individual u along a randomly chosen edge by time t. Thus, the probability that an individual u with degree k has received m pieces of behavioral information from distinct neighbors by time t can be expressed as

$$\begin{eqnarray}&&{\phi }_{m}(k,t)=\displaystyle \left(\genfrac{}{}{0em}{}{k}{m}\right){[\theta (t)]}^{k-m}{[1-\theta (t)]}^{m}.\end{eqnarray} \tag{ 1 }$$

Individual u in the susceptible state implies that the cumulative pieces of behavioral information m he received is still less than his adoption threshold T_u. According to the social contagion model in section 2, the adoption threshold and degree are independent. Considering all possible values of m and T_u, it can be obtained that the probability of individual u with degree k being susceptible is

$$\begin{eqnarray}&&s(k,t)=\displaystyle \sum _{{T}_{u}}F({T}_{u})\displaystyle \sum _{m=0}^{{T}_{u}-1}{\phi }_{m}(k,t).\end{eqnarray} \tag{ 2 }$$

Combing the degree distribution of a network, it can be known that the fraction of susceptible individuals at time step t is

$$\begin{eqnarray}&&S(t)=\displaystyle \sum _{k}P(k)s(k,t).\end{eqnarray} \tag{ 3 }$$

Once $\theta (t)$ is known, we can get S(t).

As a neighbor of individual u may be in one of the three states of susceptible, adopted or recovered, $\theta (t)$ can be divided into three cases as

$$\begin{eqnarray}&&\theta (t)={\xi }_{S}(t)+{\xi }_{A}(t)+{\xi }_{R}(t).\end{eqnarray} \tag{ 4 }$$

And ${\xi }_{S}(t)$ ${\rm{[}}{\xi }_{A}(t)$ or ${\xi }_{R}(t){\rm{]}}$ denotes the probability that a neighbor of u is in the susceptible (adopted or recovered) state and has not transmitted the behavioral information to individual u up to time t.

Then, let us explore the above three terms. If individual v with degree ${k}^{\prime }$ is susceptible initially, he can not transmit the behavioral information to u, but only receive it from other ${k}^{\prime }-1$ neighbors since u in the cavity state. Thus, the probability that individual v has received m pieces of behavioral information by time t is

$$\begin{eqnarray}&&{\tau }_{m}({k}^{\prime },t)=\displaystyle \left(\genfrac{}{}{0em}{}{{k}^{\prime }-1}{m}\right){[\theta (t)]}^{{k}^{\prime }-m-1}{[1-\theta (t)]}^{m}.\end{eqnarray} \tag{ 5 }$$

Taking all possible values of m and T_v into consideration, we can obtain that the probability of individual v remaining in susceptible state is (similar to equation (2))

$$\begin{eqnarray}&&{\rm{\Theta }}({k}^{\prime },t)=\displaystyle \sum _{{T}_{v}}F({T}_{v})\displaystyle \sum _{m=0}^{{T}_{v}-1}{\tau }_{m}({k}^{\prime },t).\end{eqnarray} \tag{ 6 }$$

The probability of an edge connecting an individual with degree ${k}^{\prime }$ is ${k}^{\prime }P({k}^{\prime })/\langle k\rangle$ for uncorrelated networks, where $\langle k\rangle$ is the mean degree. Thus, it can be obtained that

$$\begin{eqnarray}&&{\xi }_{S}(t)=\displaystyle \frac{{\displaystyle \sum }_{{k}^{\prime }}{k}^{\prime }P({k}^{\prime }){\rm{\Theta }}({k}^{\prime },t)}{\langle k\rangle }.\end{eqnarray} \tag{ 7 }$$

Subsequently, we turn to the expressions of ${\xi }_{A}(t)$ and ${\xi }_{R}(t)$. Once the behavioral information is transmitted through an edge with probability λ, the edge will no longer satisfy the definition of $\theta (t)$. Thus, the decrease of the fraction from $\theta (t)$ to $1-\theta (t)$ is $\lambda {\xi }_{A}(t)$, which can be expressed as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\theta (t)}{{\rm{d}}t}=-\lambda {\xi }_{A}(t).\end{eqnarray} \tag{ 8 }$$

For the growing of ${\xi }_{R}(t)$, two conditions must be met simultaneously. At time t, the behavioral information does not transmit through an edge with probability $1-\lambda$, and the adopted individual enters recovered state with probability γ. Then

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\xi }_{R}(t)}{{\rm{d}}t}=\gamma (1-\lambda ){\xi }_{A}(t).\end{eqnarray} \tag{ 9 }$$

Based on equations (8) and (9), and the initial conditions of $\theta (0)=1$ and ${\xi }_{R}(0)=0$, we can get the integration constant $\gamma (1-\lambda )/\lambda$ and

$$\begin{eqnarray}&&{\xi }_{R}(t)=\displaystyle \frac{\gamma [1-\theta (t)](1-\lambda )}{\lambda }.\end{eqnarray} \tag{ 10 }$$

Inserting equations (7) and (10) into (4), the following expression can be obtained that

$$\begin{eqnarray}&&{\xi }_{A}(t)=\theta (t)-\displaystyle \frac{{\displaystyle \sum }_{{k}^{\prime }}{k}^{\prime }P({k}^{\prime }){\rm{\Theta }}({k}^{\prime },t)}{\langle k\rangle }-\displaystyle \frac{\gamma [1-\theta (t)](1-\lambda )}{\lambda }.\end{eqnarray} \tag{ 11 }$$

Substituting equation (11) into (8), we can get the time evolution of $\theta (t)$ in detail

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\theta (t)}{{\rm{d}}t}=-\lambda [\theta (t)-\displaystyle \frac{{\displaystyle \sum }_{{k}^{\prime }}{k}^{\prime }P({k}^{\prime }){\rm{\Theta }}({k}^{\prime },t)}{\langle k\rangle }]+\gamma [1-\theta (t)](1-\lambda ).\end{eqnarray} \tag{ 12 }$$

Susceptible individuals move into the adopted states once they adopt the behavior. Meanwhile, the adopted individuals may lose interest in the behavior and become recovered. Thus, we can easily get the evolution of adopted and recovered individuals as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}A(t)}{{\rm{d}}t}=-\displaystyle \frac{{\rm{d}}S(t)}{{\rm{d}}t}-\gamma A(t)\end{eqnarray} \tag{ 13 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}R(t)}{{\rm{d}}t}=\gamma A(t),\end{eqnarray} \tag{ 14 }$$

respectively. Equations (1)–(3) and (12)–(14) give us a complete and general description of heterogeneous social contagions, from which the fraction in each state at arbitrary time step can be calculated. When $t\to \infty$, we can get the final adoption size $R(\infty )$.

In this subsection, we pay attention to the behavior adoption threshold with a binominal distribution F(T). More specifically, a fraction of individuals p has a relatively low adoption threshold T_a, whereas the remaining ones have a higher adoption threshold T_b. F(T) can be expressed as

$$\begin{eqnarray}F(T)=\left\{\begin{array}{ll}{T}_{a}, & \mathrm{with}\;\mathrm{probability}\;p,\\ {T}_{b}, & \mathrm{with}\;\mathrm{probability}\;1-p.\end{array}\right.\end{eqnarray} \tag{ 15 }$$

For simplicity, the values of adoption thresholds are defined as T_a  =  1 and ${T}_{b}\geqslant 1$. Individuals with low adoption threshold T_a are considered as activists, while those with high adoption threshold T_b are regarded as bigots. We herein name this kind of social contagion model as two-state spreading threshold model. And the edge-based compartmental theory is utilized to analyze the two-state spreading threshold model by substituting equation (15) into the corresponding equations, thus obtaining the value of S(t), A(t) and R(t). Particularly, we rewrite equations (2) and (6) as

$$\begin{eqnarray}&&s{(k,t)=p\theta (t)}^{k}+(1-p)\displaystyle \sum _{m=0}^{{T}_{b}-1}{\phi }_{m}(k,t),\end{eqnarray} \tag{ 16 }$$

and

$$\begin{eqnarray}&&{\rm{\Theta }}{({k}^{\prime },t)=p\theta (t)}^{{k}^{\prime }-1}+(1-p)\displaystyle \sum _{m=0}^{{T}_{b}-1}{\tau }_{m}({k}^{\prime },t),\end{eqnarray} \tag{ 17 }$$

respectively. At time t, the fractions of susceptible individuals in the activist and bigot populations are given by

$$\begin{eqnarray}&&{S}_{a}{(t)=\displaystyle \sum _{k}P(k)\theta (t)}^{k},\end{eqnarray} \tag{ 18 }$$

and

$$\begin{eqnarray}&&{S}_{b}(t)=\displaystyle \sum _{k}P(k)\displaystyle \sum _{m=0}^{{T}_{b}-1}{\phi }_{m}(k,t),\end{eqnarray} \tag{ 19 }$$

respectively. Considering the fractions of the activist and bigot populations, the density of susceptible individuals at time step t can also be written as

$$\begin{eqnarray}&&S(t)={{pS}}_{a}(t)+(1-p){S}_{b}(t).\end{eqnarray} \tag{ 20 }$$

Another issue concerned in the paper is the effects of heterogeneous adoption threshold on phase transition. To analyze the phase transition, we address the fixed point (root) of equation (12) at the steady state (i.e., $t\to \infty$) with equation (17). That is the fixed point of

$$\begin{eqnarray}&&\theta (\infty )=y[\theta (\infty )],\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&y[\theta (\infty )]=\displaystyle \frac{{\displaystyle \sum }_{{k}^{\prime }}{k}^{\prime }P({k}^{\prime }){\rm{\Theta }}({k}^{\prime },\infty )}{\langle k\rangle }+\displaystyle \frac{\gamma [1-\theta (\infty )](1-\lambda )}{\lambda }.\end{eqnarray} \tag{ 22 }$$

From figure 1(a), it can be seen that the number of nontrivial roots is 0, 1 or 3 when ${T}_{b}\geqslant 3$. For the case of ${T}_{b}=2$, the number of nontrivial roots is 0, 1 or 2 (see figure 1(b)).

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3.2.1. Case of ${T}_{b}\geqslant 3$

In this subsection, we discuss the case of ${T}_{b}\geqslant 3$. For the given P(k), p and γ, equation (21) has only one trivial solution $\theta (\infty )=1\;\mathrm{when}\;\lambda$ is small. With the increase of $\lambda ,\theta (\infty )$ decreases continuously to a nontrivial solution $\theta (\infty )\lt 1$ first (see example in figure 1(a)), which means that $R(\infty )$ grows continuously first. That is to say, there is a second-order (continuous) phase transition. By setting $\theta (\infty )$ and y $[\theta (\infty )]$ tangent at $\theta (\infty )=1$ [34, 45], we can get the continuous critical information transmission probability as

$$\begin{eqnarray}&&{\lambda }_{{\rm{c}}}^{\mathrm{II}}=\displaystyle \frac{\gamma \langle k\rangle }{p(\langle {k}^{2}\rangle -\langle k\rangle )-(1-\gamma )\langle k\rangle },\end{eqnarray} \tag{ 23 }$$

where $\langle k\rangle$ and $\langle {k}^{2}\rangle$ are the first and second moments of degree distribution, respectively. The critical value ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ separates the local behavior adoption from the global behavior adoption. A local behavior adoption means that the behavior can be adopted by few individuals, and a global behavior adoption represents that the behavior can be adopted by a finite fraction of individuals (i.e., the final adoption size is proportional to the number of individuals in the network). From equation (23), it is discovered that the emergence of global behavior adoption is determined by the network topology (i.e., degree distribution), the fraction of activists p and the recovery probability $\gamma$. The global behavior adoption is more likely to occur (i.e., a lower ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$) in scale-free networks with divergent second moment degree distribution (i.e., $\langle {k}^{2}\rangle \to \infty$). Increasing the value of p can facilitate the global behavior adoption (i.e., a lower ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$). When $p=1$, equation (23) returns to the case of epidemic outbreak threshold [43].

Fixing all the parameters except p, as similar to equation (23), we get the continuous critical fraction of activists as

$$\begin{eqnarray}&&{p}_{{\rm{c}}}^{\mathrm{II}}=\displaystyle \frac{\gamma \langle k\rangle }{\lambda (\langle {k}^{2}\rangle -\langle k\rangle )}+\displaystyle \frac{(1-\gamma )\langle k\rangle }{\langle {k}^{2}\rangle -\langle k\rangle }.\end{eqnarray} \tag{ 24 }$$

From equation (24), it can be known that an enough fraction of activists are necessary for triggering the global behavior adoption, and ${p}_{{\rm{c}}}^{\mathrm{II}}$ decreases with the increase of network heterogeneity, p and $\lambda$. By setting $\gamma =0$ in equation (24), another critical proportion of activists ${p}_{{\rm{c}}}^{\star }$ can be obtained, below which any values of $\lambda$ can not trigger the global behavior adoption, and it is

$$\begin{eqnarray}&&{p}_{{\rm{c}}}^{\star }=\displaystyle \frac{\langle k\rangle }{\langle {k}^{2}\rangle -\langle k\rangle }.\end{eqnarray} \tag{ 25 }$$

It is worth noting that equation (25) is the same with network's percolation condition [45], which means that the global behavior adoption is possible only when the activists percolate the entire network (i.e., activists can form a finite connected cluster).

As shown in figure 1(a), three nontrivial roots of equation (21) occur when $\lambda$ is large enough (see figure 1(a) for $\lambda =0.615$). This phenomenon is caused by the bigots, since equation (21) has at most one nontrivial root when the bigots are absent [26]. In this case, the meaningful solution will be given by the largest stably root (since only this value can be achieved physically). For $\lambda ={\lambda }_{{\rm{c}}}^{{\rm{I}}}=0.635$, the tangent point is the solution. For $\lambda \gt {\lambda }_{{\rm{c}}}^{{\rm{I}}}$, the meaningful solution is the only stably fixed point. The meaningful solution of equation (21) changes abruptly to a small value from a relatively large value at ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ (see the insert figure 1(a)), resulting in a discontinuous growth of $R(\infty )$. Based on the bifurcation theory [46], the discontinuous critical information transmission probability can be obtained as follows

$$\begin{eqnarray}&&{\lambda }_{{\rm{c}}}^{{\rm{I}}}=\displaystyle \frac{\gamma }{{\rm{\Delta }}+\gamma -1},\end{eqnarray} \tag{ 26 }$$

where

$$\begin{eqnarray*}&&{\rm{\Delta }}=\displaystyle \frac{{\displaystyle \sum }_{{k}^{\prime }}{k}^{\prime }P({k}^{\prime })\frac{{\rm{d}}{\rm{\Theta }}({k}^{\prime },\infty )}{{\rm{d}}\theta (\infty )}{| }_{{\theta }_{s}(\infty )}}{\langle k\rangle },\end{eqnarray*}$$

and ${\theta }_{s}(\infty )$ is the fixed point of equation (21). Combining equations (5) and (17), we can get that

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\rm{\Theta }}({k}^{\prime },\infty )}{{\rm{d}}\theta (t)}=p({k}^{\prime }-1)\theta {(\infty )}^{{k}^{\prime }-2}+(1-p){ \mathcal X },\end{eqnarray} \tag{ 27 }$$

where

$$\begin{eqnarray*}&&{ \mathcal X }=\displaystyle \sum _{m=0}^{{T}_{b}-1}\displaystyle \left(\genfrac{}{}{0em}{}{{k}^{\prime }-1}{m}\right)\{({k}^{\prime }-m-1)\theta {(\infty )}^{{k}^{\prime }-m-2}{[1-\theta (\infty )]}^{m}-m\theta {(\infty )}^{{k}^{\prime }-m-1}{[1-\theta (\infty )]}^{m-1}\}.\end{eqnarray*}$$

Using the analytical method similar to equation (26), it is obtained that the discontinuous critical fraction of activists can be expressed as

$$\begin{eqnarray}&&{p}_{{\rm{c}}}^{I}=\displaystyle \frac{\lambda (1-{ \mathcal Y })+\gamma (1-\lambda )}{\lambda ({ \mathcal Z }-{ \mathcal Y })},\end{eqnarray} \tag{ 28 }$$

where

$$\begin{eqnarray*}&&{ \mathcal Y }=\displaystyle \frac{{\displaystyle \sum }_{{k}^{\prime }}{k}^{\prime }P({k}^{\prime }){ \mathcal X }}{\langle k\rangle }\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&{ \mathcal Z }=\displaystyle \frac{{\displaystyle \sum }_{{k}^{\prime }}{k}^{\prime }({k}^{\prime }-1)P({k}^{\prime }){\theta }^{{k}^{\prime }-2}}{\langle k\rangle }.\end{eqnarray*}$$

From the above analysis, we find that $R(\infty )$ versus $\lambda$ or p first grows continuously and then follows a discontinuous fashion. And the continuous and discontinuous growthes of $R(\infty )$ are caused by the activists and bigots, respectively, which can be regarded as hybrid phase transition [47] from the perspective of statistical physics, due to its mixture of the traditional first-order and second-order transitions. Note that the hybrid phase transition can change to a second-order phase transition. Numerically solving equations (21) and (26), and the following equation

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}}^{2}y[\theta (\infty )]}{{\rm{d}}{\theta }^{2}(\infty )}=0,\end{eqnarray} \tag{ 29 }$$

we can learn the condition under which the hybrid phase transition disappears.

3.2.2. Case of ${T}_{b}=2$

To be illustrative, we first focus on random regular networks (RRNs). Figure 2(a) shows the time evolution of the fraction of adopted individuals ${A}_{a}(t)$ and ${A}_{b}(t)$ in the activist and bigot populations with different behavioral information transmission probabilities $\lambda$. It is found that a hierarchical characteristic of behavior adoption is caused by heterogeneous adoption thresholds. That is to say, activists with low ${T}_{a}$ first adopt the behavior and then stimulate the bigots with ${T}_{b}$ to adopt the behavior. With a relatively small $\lambda =0.5,{A}_{a}(t)$ shows a small peak, and can not stimulate many bigots to adopt the behavior (${A}_{b}(t)$ does not see an obvious peak). With a relatively large $\lambda =0.7,{A}_{a}(t)$ shows a large peak, and further leads to the emergence of a large peak for ${A}_{a}(t)$. The heterogeneous adoption threshold distribution may be used to understand some real-world hierarchical phenomena of behavior adoption, such as the adoption of Skype serves [38]. The time evolution can be well predicted by our edge-based compartmental theory.

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From figure 2(b), it can be seen that $R(\infty )$ versus $\lambda$ shows a hybrid phase transition, which means that $R(\infty )$ first grows continuously and then follows a discontinuous fashion. The continuous and discontinuous phase transitions are caused by activists and bigots, respectively. Similar to [26], the discontinuous growth of $R(\infty )$ is caused by those bigots in the subcritical state who adopt the behavior simultaneously. An individual in such a state has received the behavioral information but has not yet adopted the behavior, and the pieces of information from distinct neighbors is precisely one less than his adoption threshold. The theoretical (numerical) values of ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ and ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ separate figures 2(b) and (c) into three regions. The theoretical values of ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ and ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ can be gotten from equations (23) and (26), respectively. In region I, with $\lambda \leqslant {\lambda }_{{\rm{c}}}^{\mathrm{II}}$, both activist and bigot populations adopt the behavior locally (i.e., only a vanishingly small fraction of individuals adopted the behavior). In region II, with ${\lambda }_{{\rm{c}}}^{\mathrm{II}}\lt \lambda \leqslant {\lambda }_{{\rm{c}}}^{{\rm{I}}}$, activists adopt the behavior globally (i.e., a finite fraction of activists adopted the behavior) and bigots adopt the behavior locally. In region III with $\lambda \gt {\lambda }_{{\rm{c}}}^{{\rm{I}}}$, both the activists and bigots adopt the behavior globally. The numerical values of ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ and ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ can be obtained by observing ${v}_{R}$ in figure 2(c). For instance, ${v}_{R}$ has two peaks, which means that two phase transitions occur [48]. And the first peak appears at ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$, while the second peak locates at ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$. Note that the first (second) peak of ${v}_{R}$ shares the same location with the maximal peak of activist population ${v}_{R}^{a}$ (bigots population ${v}_{R}^{b}$). In all, ${\lambda }_{{\rm{c}}}^{{\rm{I}}},{\lambda }_{{\rm{c}}}^{\mathrm{II}}$ and $R(\infty )$ can be well predicted by our edge-based compartmental theory.

As shown in figure 3, $R(\infty )$ and the phase transition are significantly influenced by the adoption threshold of bigots ${T}_{b}$. And $R(\infty )$ decreases with ${T}_{b}$, since a larger value of ${T}_{b}$ means the bigots need to be exposed to more information before adopting the behavior. The phase transition is continuous when ${T}_{b}=1$. In this case, our model reduces to the SIR model [8]. The phase transition is also continuous when ${T}_{b}\geqslant 6$, since there are not enough activists to persuade bigots to adopt the behavior simultaneously. For the case of ${T}_{b}=2,R(\infty )$ shows a first-order phase transition, because the bigots are likely to enter subcritical states and adopt the behavior simultaneously. A hybrid phase transition emerges with other values of ${T}_{b}$ (i.e., $2\lt {T}_{b}\lt 6$). As discussed in section 3.2, the type of phase transition is verified by bifurcation analysis of equation (21). As shown in figure 3(b), the simulated values of ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ and ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ are located by studying ${v}_{R}$ versus $\lambda$. For the second-order phase transition, ${v}_{R}$ has only one peak (see ${T}_{b}=1$ and ${T}_{b}=6$ in figure 3(b)). Similarly, ${v}_{R}$ also has only one peak for the first-order phase transition (see ${T}_{b}=2$ in figure 3(b)). For the hybrid phase transition, ${v}_{R}$ has two peaks (see $3\leqslant {T}_{b}\leqslant 5$ in figure 3(b)). Our theoretical predictions of $R(\infty )$ agree well with the simulation results, except for $\lambda$ close to the critical information transmission probability. The deviations between our predictions and simulation are mainly derived from the finite-size effects of networks, as shown in figure 4. The deviations of $R(\infty ),{\lambda }_{{\rm{c}}}^{I}$ and ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ between the simulated and theoretical results decrease with network size $N$.

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Then, we observe $R(\infty )$ versus p for different $\lambda$ in figure 5. We find that $R(\infty )$ increases with p. By bifurcation analysis of equation (21), a continuous growth of $R(\infty )$ is observed with a relatively small $\lambda$ (e.g., $\lambda =0.5$), and the hybrid phase transition occurs with a relatively large $\lambda$ (e.g., $\lambda =0.7$ and 0.8 ). The numerical values of ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ and ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ are located by studying ${v}_{R}$ (see figure 5(b)). Again, the theoretical and numerical results agree well.

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From the above analysis, it can be obtained that both $\lambda$ and p markedly affect $R(\infty )$ and phase transition. Thus, $R(\infty )$ and the phase transition on parameter plane $(\lambda ,p)\;\mathrm{when}\;{T}_{b}=4$ are further investigated in figure 6. Obviously, $R(\infty )$ increases with $\lambda$ and p. According to the type of phase transition, the parameter plane $(\lambda ,p)$ is divided into four different regions by three vertical lines. The first vertical line can be gotten from equation (25), and the other two can be predicted by solving equations (21), (26) and (29). In region I ($p\leqslant {p}_{{\rm{c}}}^{\star }=1/9$), there are a few activists, who can not percolate the entire population. Thus, no matter what the value of $\lambda$ is, activists can not be made to adopt the behavior globally. When $p\gt {p}_{{\rm{c}}}^{\star }$, the global behavior adoption becomes possible, and a crossover phenomenon, which means that the phase transition changes from being hybrid to being second-order, occurs in the phase transition. Meanwhile, the local and global behavior adoptions are separated by the red solid curve (i.e., ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$). In region II ($1/9\lt p\leqslant 0.15$), the relatively few activists lead to the continuous phase transition. In this region, $R(\infty )$ grows continuously versus $\lambda$ for a given p, and a finite fraction of individuals adopt the behavior above ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$. With the increase of p, in region III ($0.15\lt p\leqslant 0.5$), the hybrid phase transition occurs, i.e., $R(\infty )$ first grows continuously with $\lambda$ and then follows by a discontinuous pattern. A finite fraction of activists adopt the behavior above ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$, and further induce the bigots to adopt the behavior simultaneously above ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ (see black curves obtained from equation (26)). In region IV ($p\gt 0.5$), half of the neighbors of bigots are activists. Once these activists adopt the behavior, the bigots will gradually adopt the behavior. Thus, $R(\infty )$ grows continuously and a finite fraction of individuals adopt the behavior above ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$ (see red curves). Our theoretical predictions of ${\lambda }_{{\rm{c}}}^{\mathrm{II}},{\lambda }_{{\rm{c}}}^{I}$ and $R(\infty )$ agree well with the numerical predictions.

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It can be seen in figure 3 that $R(\infty )$ increases discontinuously with $\lambda \quad \mathrm{when}\quad {T}_{b}=2$, thus $R(\infty )$ and phase transition on parameter plane p$\lambda$ when ${T}_{b}=2$ is explored in figure 7. And we find another crossover phenomenon in the phase transition: a change from being first-order to being second-order. Here, what is similar to figure 6 is that the plane is divided into three regions: region I ($p\leqslant 1/9$), the local behavior adoption region, in which only a vanishingly small fraction of individuals adopt the behavior; region II ($1/9\lt p\leqslant 0.42$) shows a first-order phase transition, where a finite fraction of individuals adopt the behavior simultaneously above ${\lambda }_{{\rm{c}}}^{{\rm{I}}}$ (red dished lines); region III ($p\gt 0.42$) exhibits a second-order phase transition, in which $R(\infty )$ increases continuously versus $\lambda$. The type of phase transition is verified by bifurcation analysis.

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We turn to investigate the effects of network topological parameters, including the mean degree and degree heterogeneity, on the two-state spreading threshold model. We first examine RRNs with different values of mean degree $\langle k\rangle$. As shown in figure 8, the behavior spreads easier for a greater value of $\langle k\rangle$. For the case of ${T}_{b}=2$, the type of phase transition is not altered by $\langle k\rangle$, i.e., $R(\infty )$ always increases discontinuously with $\lambda$ (see figure 8(a)). For the case of ${T}_{b}=4$ in figure 8(b), the phase transition changes from a hybrid type for a relatively large mean degree (e.g., $\langle k\rangle =10$) to a continuous type for a small value of $\langle k\rangle =5$. We verify the type of phase transition by bifurcation analysis. The numerical critical values are located by studying ${v}_{R}$ in figures 8(c) and (d). The theory can predict simulation results $R(\infty )$ very well except for $\lambda$ close to the critical points. The deviations are induced by the finite-size effects of networks.

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We next study the effects of network heterogeneity. To build the heterogeneous networks, the uncorrelated configuration model with power-law degree distributions $P(k)\sim {k}^{-{\gamma }_{D}}$ is used, where the mean degree $\langle k\rangle =10$ and the maximum degree ${k}_{\mathrm{max}}\sim \sqrt{N}$ [39]. The network heterogeneity increases with the decrease of ${\gamma }_{D}$.

In the case of ${T}_{b}=4$, we find that the global behavior adoption more likely to occur (i.e., lower ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$) in heterogeneous networks, due to the existence of hubs in heterogeneous networks [49], as shown in figure 9. Meanwhile, in strong heterogeneous networks a large number of individuals with small degrees are difficult to adopt the behavior, so $R(\infty )$ is smaller at large $\lambda$. For example, $R(\infty )$ at $\lambda =0.7$ is obviously smaller on scale-free networks with ${\gamma }_{D}=2.5$ than that on RRNs. By bifurcation analysis of equation (21), it is discovered that the hybrid phase transition disappears for strong heterogeneous networks (e.g., ${\gamma }_{D}=2.5\;\;\mathrm{and}\;\;3.0$ in figure 9). That is to say, network heterogeneity leads to the crossover phenomenon: a change from hybrid to second-order phase transition. Moreover, the numerical critical values are further located by observing ${v}_{R}$ in figure 9(b). Evidences in terms of the quantities $R(\infty ),{\lambda }_{{\rm{c}}}^{I}$ and ${\lambda }_{{\rm{c}}}^{\mathrm{II}}$, which support our edge-based compartmental theory.

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For the case of ${T}_{b}=2$ (see figure 10), we find the similar phenomenon of final behavior adoption size $R(\infty )$ increases (decreases) with network heterogeneity when $\lambda$ is small (large). Based on bifurcation theory, we find that the system has the first-order phase transition, which denotes that network heterogeneity does not alter the phase transition. As shown in figure 10(b), we locate the numerical critical values by studying ${v}_{R}$. The deviations between the theoretical predictions and simulation results are mainly caused by the finite-size effects of networks.

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