Impact of contact preference on social contagions on complex networks

Preferential contact process limited by contact capacity remarkably affects the spreading dynamics on complex networks, but the influence of this preferential contact in social contagions has not been fully explored. To this end, we propose a behavior spreading model based on the mechanism of preferential contact. The probability in the model that an adopted individual contacts and tries to transmit the behavioral information to one of his/her neighbors depends on the neighbor's degree. Besides, a preferential exponent determines the tendency to contact with either small-degree or large-degree nodes. We use a dynamic messaging method to describe this complex contagion process and verify that the method is accurate to predict the spreading dynamics by numerical simulations on strongly heterogeneous networks. We find that the preferential contact mechanism leads to a crossover phenomenon in the growth of final adoption size. By reducing the preferential exponent, we observe a change from a continuous growth to an explosive growth and then to a continuous growth with the transmission rate of behavioral information. Moreover, we find that there is an optimal preferential exponent which maximizes the final adoption size at a fixed information transmission rate, and this optimal preferential exponent decreases with the information transmission rate. The used theory can be extended to other types of dynamics, and our findings provide useful and general insights into social contagion processes in the real world.

Figure 1

Effects of preferential contacts on simple contagions. (a) The final adoption size $R\left(\infty \right)$ versus the information transmission rate $\lambda$ for different preferential exponents $\alpha$ when $A_0=0.1$, where green up triangles, blue squares, red circles, black diamonds, and mauve asterisks represent the simulation results for $\alpha =2,1,0,-1,-2$, respectively. (b) $R\left(\infty \right)$ versus $\alpha$ for different $\lambda$ values when $A_0=0.1$, where red circles, blue squares, and green up triangles represent the simulation results for $\lambda =0.2,0.1,0.06$, respectively. (c) $R\left(\infty \right)$ versus $\lambda$ for different $\alpha$ when $A_0=0.0001$. (d) The epidemic threshold ${\lambda }_c$ versus $\alpha$ when $A_0=0.0001$ which is different from $A_0=0.1$ in panels (a) and (b), where the red circles and blue squares represent the numerical thresholds identified by the variability measure for $\nu =4.0,2.1$. In panels (a), (b), and (c), the curves are the theoretical predictions from Eqs. (2, 3, 4), (7), and (15). In panel (d), the curve is the theoretical prediction from Eq. (20). Other parameters are $\nu =2.1,c=4,\kappa =1$, and $\gamma =0.1$.

Figure 2

Effects of preferential contacts on complex contagions. (a) $R\left(\infty \right)$ as a function of $\lambda$ for different values of $\alpha$, where green up triangles, blue squares, red circles, black diamonds, and mauve asterisks represent the simulation results for $\alpha =2,1,0,-1,-2$, respectively. (b) The final density of subcritical state $\mathrm{\Phi }(\kappa -1,\infty )$ as a function of $\lambda$ for different values of $\alpha$, where red circles and black squares are the simulation results for $\alpha =1,-1$, respectively. (c) $R\left(\infty \right)$ as a function of $\alpha$ for different $\lambda$ values, where red circles, blue squares, and green up triangles represent the simulation results for $\lambda =0.48,0.28,0.18$, respectively. In all panels, the curves are the theoretical predictions from Eqs. (2, 3, 4), (7), and (15). Other parameters are $\nu =2.1,c=4,\kappa =3$, and $\gamma =0.1$.

Figure 3

Comparison of theoretical critical points with numerical critical points of complex contagions on a scale-free network. (a) The final adoption size $R\left(\infty \right)$ with error bar as a function of information transmission rate $\lambda$. (b) The variability $\chi$ (red circles) and the jump size $\mathrm{\Delta }R(\lambda ,\infty )$ (blue squares) versus $\lambda$. Other parameters are $\nu =2.1,c=4,\kappa =3,\gamma =0.1$, and $\alpha =-1$.

Figure 4

Phase diagram of the final adoption size in the two-parameter $(\lambda ,\alpha )$ space. (a) The color-coded values of $R\left(\infty \right)$ are from numerical simulations and (b) the color-coded values of $R\left(\infty \right)$ are from theoretical predictions. In all panels, the horizontal white lines indicate the critical preferential exponents ${\alpha }_c^d\approx -1.8$ and ${\alpha }_c^u\approx -0.6$, respectively. White circles and red dashed curve, respectively, represent the numerical and the theoretical ${\lambda }_c$ values. The yellow solid curves and yellow dashed curves are numerical and the theoretical results of ${\alpha }_o$. Other parameters are $\nu =2.1,c=4,\kappa =3$, and $\gamma =0.1$.

Figure 5

Effects of preferential contacts on complex contagions on U. Rovira i Virgili network and Hamsterster friendships network. $R\left(\infty \right)$ as a function of $\lambda$ for different values of $\alpha$ on U. Rovira i Virgili network (a) and Hamsterster friendships network (d), where green up triangles, red circles, and mauve asterisks represent the simulation results for $\alpha =2.0,-1.0,-3.0$, respectively. $R\left(\infty \right)$ as a function of $\alpha$ for different $\lambda$ values on U. Rovira i Virgili network (b) and Hamsterster friendships network (e), where red circles, blue squares, and green up triangles represent the simulation results for $\lambda =0.48,0.28,0.18$, respectively. The explosive growth of the final adoption size $R\left(\infty \right)$ with the information transmission rate $\lambda$ on U. Rovira i Virgili network (c) and Hamsterster friendships network (f), where each circle represents the result of one dynamical realization. In all panels, the curves are the theoretical predictions from Eqs. (2, 3, 4), (7), and (15). Other parameters are $c=8,\kappa =3,A_0=0.05,$ and $\gamma =0.1$.

Figure 6

Effects of network size on the accuracy of the dynamic message-passing method. $R\left(\infty \right)$ as a function of $\lambda$ for different values of $N$, where the green up triangles, blue squares and red circle, respectively, represent the simulation results for $N=1000,5000,10000$. And the green solid, blue dotted and red dot-dashed curves represent the theoretical predictions for $N=1000,5000,10000$ from Eqs. (2, 3, 4), (7), and (15), respectively. Other parameters are $\nu =2.1,c=4,\kappa =3,A_0=0.1,$ and $\gamma =0.1$.

Figure 7

Effects of network size on the behavior spreading. Other parameters are $\nu =2.1,c=4,\kappa =3,A_0=0.1$, and $\gamma =0.1$.

Figure 8

Effects of mean degree on the behavior spreading. Other parameters are $\nu =2.1,c=4,\kappa =3,A_0=0.1,$ and $\gamma =0.1$.