Social contagion induced by uncertain information

Information and individual activities often spread globally through the network of social ties. While social contagion phenomena have been extensively studied within the framework of threshold models, it is common to make an assumption that may be violated in reality: each individual can observe the neighbors' states without error. Here, we analyze the dynamics of global cascades under uncertainty in an otherwise standard threshold model. Each individual uses statistical inference to estimate the probability distribution of the number of active neighbors when deciding whether to be active, which gives a probabilistic threshold rule. Unlike the deterministic threshold model, the spreading process is generally nonmonotonic, as the inferred distribution of neighbors' states may be updated as a new signal arrives. We find that social contagion may occur as a self-fulfilling event in that misperception may trigger a cascade in regions where cascades would never occur under certainty.

Figure 1

Schematic of Bayesian inference and node activation. An individual does not have complete information about the true number of active neighbors $m$ and the fraction of active neighbors $\overline{q}$. The individual receives a signal $\tilde{m}$ that follows a binomial distribution $B(k,\overline{q})$. Conditional on the realized signal, the distribution of the true fraction of active neighbors [denoted by $f\left(q\right|\tilde{m})$] is inferred. The individual is activated if $F\left(\theta \right|\tilde{m})<\lambda$, where $F\left(\theta \right|\tilde{m})={\int }_0^{\theta }f\left(q\right|\tilde{m})dq$.

Figure 2

Prior and posterior distributions. (a) Prior (dotted) and posterior (black solid) distributions are more concentrated around the true fraction of active nodes $\overline{q}$ (vertical dotted line) when $\beta$ is larger. (b) Complementary cumulative distribution function (CCDF) of the posterior distribution, $F\left(q\right|\tilde{m})$. ${\overline{\theta }}_1$ (${\overline{\theta }}_2$) denotes the threshold of $\theta$ below which the node is activated for a given accommodation level ${\lambda }_1$ (${\lambda }_2$). We have ${\theta }_1\approx {\theta }_2$ when $\beta$ is sufficiently large. We set $k=10, \overline{q}=0.4$, and $\tilde{m}=4$.

Figure 3

Phase diagram and cascade region. (a) Phase diagram based on the MF equation (15) with $\lambda =0.1$ and $\lambda =0.7$. Red circle indicates ${\rho }^1={\rho }_0=0.01$. $\theta =0.18, \beta =20$, and $\langle k\rangle =8$. (b) Cascade region obtained by simulation (color) and the cascade conditions (lines). The areas surrounded by red solid, red dashed, and black solid lines denote the cascade regions indicated by the AMEs [i.e., ${\rho }_1\left(T\right)>0.1$], the sign condition (16), and the first-order cascade condition (17), respectively. We set $N={10}^5$ and $\beta =20$.

Figure 4

Shrinkage and expansion of cascade region induced by misperception. The heat maps represent the simulated fraction of active nodes for (a) $\lambda =0.1$ and (b) $\lambda =0.8$. Area surrounded by the blue dotted (red solid) line represents the difference from the cascade region under certainty, $\tilde{\mathcal{C}}\setminus \mathcal{C}$ ($\mathcal{C}\setminus \tilde{\mathcal{C}}$), indicated by the sign condition (16). $N={10}^5, \theta =0.18$, and ${\rho }_0=0.01$. Simulation results show the average over 100 runs.

Figure 5

Effects of varying the accommodation threshold $\lambda$. (a) The largest unstable solution of Eq. (15) (i.e., the threshold of ${\rho }_0$ above which a global cascade occurs). Blue and black lines, respectively, denote the values of ${\underline{\rho }}^1$ obtained by the models with and without uncertainty. ${\lambda }_{\mathrm{n}}$ denotes the neutral level of accommodation at which the effect of uncertainty is offset. (b) Theoretical (black line) and simulated (black circle) fraction of active nodes. Red cross denotes the normalized cumulative fraction of false-active nodes, $\overline{\mathrm{\Theta }}$ (right axis), which takes the maximum at $\tilde{\lambda }$. Blue triangle shows the simulated fraction of active nodes that would be attained without false-active nodes. We set $N={10}^5, \langle k\rangle =8, \theta =0.18, \beta =100$, and ${\rho }_0=0.01$. Simulation results show the average over 100 runs.

Figure 6

Steady-state fraction of active nodes and the cumulative fraction of nodes that reverted the states, $\mathcal{R}$. We set $N={10}^4, \langle k\rangle =8, \theta =0.18, \beta =5, {\rho }_0=0.01$. Average is taken over $10000$ runs.

Figure 7

Simulated cascade size under sequential Bayesian updating. The initial prior $f\left(q\right)$ is given by $\mathrm{Beta}(a,b)$, where $a=b{\rho }_0/(1-{\rho }_0)$. See the caption of Fig. 5 for the parameter settings.