Dynamics of social contagions with memory of nonredundant information

A key ingredient in social contagion dynamics is reinforcement, as adopting a certain social behavior requires verification of its credibility and legitimacy. Memory of nonredundant information plays an important role in reinforcement, which so far has eluded theoretical analysis. We first propose a general social contagion model with reinforcement derived from nonredundant information memory. Then, we develop a unified edge-based compartmental theory to analyze this model, and a remarkable agreement with numerics is obtained on some specific models. We use a spreading threshold model as a specific example to understand the memory effect, in which each individual adopts a social behavior only when the cumulative pieces of information that the individual received from his or her neighbors exceeds an adoption threshold. Through analysis and numerical simulations, we find that the memory characteristic markedly affects the dynamics as quantified by the final adoption size. Strikingly, we uncover a transition phenomenon in which the dependence of the final adoption size on some key parameters, such as the transmission probability, can change from being discontinuous to being continuous. The transition can be triggered by proper parameters and structural perturbations to the system, such as decreasing individuals' adoption threshold, increasing initial seed size, or enhancing the network heterogeneity.

Figure 1

Illustration of the susceptible-adopted-recovered (SAR) model on complex networks. (a) At time $t$, the adopted individual 5 tries to transmit the behavioral information (or simply information for short) to each susceptible neighbor individual independently with probability $\lambda$. Note that individual 5 cannot transmit the information to individual 4, since he or she has transmitted the information to individual 4 successfully before time $t$. That is to say, the susceptible individual can only get the nonredundant information from his or her neighbors. The solid blue lines denote that the information has not transmitted through them successfully, and the red dashed line denotes that the information has transmitted through it previously. (b) Assuming that individual 1 has received a new piece of information at time $t$, whether individual 1 adopts the behavior is determined by the $m$ cumulative pieces of information he or she ever received from neighbors. The value of $m$ can be expressed as $m={\sum }_{d=1}^t{m}_d$, where $m_d$ is the pieces of information that individual 1 received at time $d$. In such a situation, individual 1 has to remember the pieces of nonredundant information he or she received from neighbors before time $t$. Thus, the so-called nonredundant information memory is induced. Individual 1 becomes adopted with probability $\pi (k,m)$, where $k$ is the degree of individual 1; otherwise, individual 1 remains in the susceptible state.

Figure 2

Illustration of graphical solutions of Eq. (15). For ER random networks, (a) continuously increasing behavior of $R\left(\infty \right)$ with $\lambda$ for $T=1$, (b) discontinuous change in $R\left(\infty \right)$ for $T=3$. The black solid lines are the horizontal axis and the red dots denote the tangent points. Other parameters are ${\rho }_0=0.1$ and $\gamma =1.0$.

Figure 3

Spreading threshold model on ER networks. (a) Average densities of susceptible, adopted, and recovered populations, denoted by $S\left(t\right)$ (black squares), $A\left(t\right)$ (blue up triangles), and $R\left(t\right)$ (red circles), respectively, versus time. (b) Final behavior adoption size $R\left(\infty \right)$ versus the transmission probability $\lambda$ for $T=1$ (black squares), $T=2$ (blue up triangles), $T=3$ (red circles), $T=4$ (dark green diamonds), and $T=5$ (light green stars) in the steady state. (c) Simulation results of NOI (number of iterations) as a function of $\lambda$ with $T=2$ (blue solid line), $T=3$ (red dashed line), and $T=4$ (dark dash dotted green line). The error bars indicate the standard deviations. The lines in (a) and (b) are theoretical predictions based on solutions of Eqs. (1, 2, 3) and (12)–(14). In (a), we set $\lambda =0.8,{\rho }_0=0.1,T=3$, and $\gamma =0.5$ (so as to obtain longer evolution time), while in (b) and (c), we set ${\rho }_0=0.1$ and $\gamma =1.0$.

Figure 4

The final fraction of individuals in the subcritical state on ER networks. $\mathrm{\Phi }(T-1,\infty )$ versus $\lambda$ for $T=2$ (black squares), $T=3$ (blue up triangles), $T=4$ (red circles), and $T=5$ (dark green diamonds). The lines are theoretical predictions based on solutions of Eqs. (1), (4) and (12, 13, 14). Other parameters are $\gamma =1.0$ and ${\rho }_0=0.1$.

Figure 5

Dependence of final behavior adoption size on initial seed size and transmission probability. For spreading threshold model on ER random networks, color-coded values of $R\left(\infty \right)$ from numerical simulations (a) and theoretical solutions (b) in the parameter plane $({\rho }_0,\lambda )$, where the theoretical values are from solutions of Eqs. (1, 2, 3) and (12)–(14). The numerically obtained critical values of the transmission probability, ${\lambda }_c^I$ (white circles), are from the NOI method, and the corresponding theoretical values blue dashed line) are from Eqs. (15) and (17, 18). In each subfigure, three regions are shown: only a vanishingly small fraction of individuals can be exposed to adopt the behavior in region I, in region II $R\left(\infty \right)$ grows discontinuously with $\lambda$ and a finite fraction of individuals adopt the behavior above ${\lambda }_c^I$, and $R\left(\infty \right)$ grows continuously to a large value in region III. The vertical white solid lines and dash dotted yellow lines separate the plane into the three regions, which are predicted from our edge-based compartmental theory. Other parameters are $\gamma =1.0$ and $T=3$.

Figure 6

Effect of mean network degree on social contagion dynamics. For ER random networks, $R\left(\infty \right)$ versus the transmission probability $\lambda$ for mean degree $\langle k\rangle =5$ (gray squares), $\langle k\rangle =8$ (blue up triangles), $\langle k\rangle =10$ (red circles), $\langle k\rangle =15$ (dark green diamonds), and $\langle k\rangle =20$ (light green stars). Other parameters are ${\rho }_0=0.1,\gamma =1.0$, and $T=3$. The lines are theoretical values of $R\left(\infty \right)$ calculated from Eqs. (1, 2, 3) and (12)–(14).

Figure 7

Effect of network heterogeneity on social contagion dynamics. For scale-free networks, $R\left(\infty \right)$ versus $\lambda$ for degree exponent $\nu =2.2$ (gray squares), $\nu =3.0$ (blue up triangles), $\nu =4.0$ (red circles), and $\nu \to \infty$ (dark green diamonds). The case for $\nu \to \infty$ reduces to a random regular network with identical degree. The lines are theoretical values of $R\left(\infty \right)$ calculated from Eqs. (1, 2, 3) and (12)–(14). Other parameters are ${\rho }_0=0.1,\gamma =1.0$, and $T=3$.

Figure 8

Effect of degree-correlated spreading threshold on social contagion dynamics. For ER random networks, final adoption size $R\left(\infty \right)$ versus the transmission probability $\lambda$ for $\alpha =-2$ (gray squares), $\alpha =-1$ (blue up triangles), $\alpha =0$ (red circles), $\alpha =1$ (dark green diamonds), and $\alpha =2$ (yellow left triangles). Other parameters are $\langle k\rangle =10,\langle T\rangle =3$, and $\gamma =1.0$. The lines are theoretical values of $R\left(\infty \right)$ from solutions of Eqs. (1, 2, 3) and (12)–(14) with adoption threshold given by Eq. (29).

Figure 9

Results from a generalized social contagion model. For ER networks, $R\left(\infty \right)$ versus the unit adoption probability $\epsilon$ for $\lambda =0.3$ (a) and the transmission probability $\lambda$ for $\epsilon =0.3$ (b). Two values of ${\rho }_0$ are used: ${\rho }_0=0.01$ (gray squares) and ${\rho }_0=0.10$ (blue up triangles). Additional parameters are $\gamma =1.0$ and $\langle k\rangle =10$. In both panels, the lines represent the theoretical values of $R\left(\infty \right)$ obtained from solutions of Eqs. (1, 2, 3) and (12)–(14) with $\psi \left(t\right)$ given by Eq. (33).