Social contagion with degree-dependent thresholds

We investigate opinion spreading by a threshold model in a situation in which the influence of people is heterogeneously distributed. We assume that there is a coupling between the influence of an individual (measured by the out-degree) and the threshold for accepting a new opinion or habit. We find that if the coupling is strongly positive, the final state of the system will be a mix of different opinions. Otherwise, it will converge to a consensus state. This phenomenon cannot simply be explained as a phase transition, but it is a combined effect of mechanisms and their relative dominance in different regions of parameter space.

Figure 1

(a) A network example with $N=10,M=3$. The sizes of nodes are proportional to the number of the out-degree (their shades also reflect their out-degree). The thickness of the edges represents the number of links between nodes. (b) Threshold ${\varphi }_i$ as a function of the rank $r_i$ of the out-degree for different coupling parameters $\beta$.

Figure 2

Schematic description of the updating rule. ${\varphi }_i$ and ${\varphi }_j$ are thresholds of nodes $i$ and $j$. $t$ is the time step for one flipping of a node. Here, $M=4$ for simplicity. Each arrow indicates the direction of influence. In particular, the black thick arrows represent the in-degrees from an omitted network. The number inside the thick arrow shows the number of in-degrees. After the comparison process of the algorithm, the opinion starts to spread from $z$ at $t=1$. Finally, node $i$ changes its opinion 0 (white) to opinion 1 (gray) at $t=3$.

Figure 3

The average opinion in steady state (${\rho }_s$) (a) as a function of $p$ with different $\beta$, (b) for $N\in [200,12800]$, and (c) with different sizes to show the transition behavior around $\beta =0.4$ ($n=2000$ for $N\in [200,3200]$ and $n=200$ for $N=6400,12800$ to reduce the simulation time; standard error is shown).

Figure 4

Normalized fixation time $\left[{\tilde{\tau }}_F\left(i\right)\right]$ and final opinion $\left[o_F\left(i\right)\right]$ for different coupling strengths $\beta$ as functions of the rank $r_i$ of the nodes. Panel (a) shows $\beta =-0.1$, (b) $\beta =-0.4$, (c) $\beta =-1$, (d) $\beta =0.1$, (e) $\beta =0.4$, and (f) $\beta =1$. The arrow in panel (e) points to the most influential node, which is also the first to fixate its opinion. Each ${\tilde{\tau }}_F\left(i\right)$ and $o_F\left(i\right)$ averaged for 2000 runs ($n=2000$) with $N=800$.

Figure 5

Panel (a) displays the average opinion as a function of time $t$ measured in time steps with $N=800$. The dashed black line is our semianalytic calculation for the initial change (integrating the equations by the Runge-Kutta method). ${\rho }_s$ is the final average opinion at the steady state of an entire network (error bars show the standard error). Panel (b) shows the difference between average opinions based on the internal mean field at the steady state of the initiator group ($\rho \left[{\tilde{\tau }}_F\left(\widehat{G}\right)\right]$) and maximum threshold (${\varphi }_{\max }$). $\beta =0.4$ is the first point that $\rho \left[{\tilde{\tau }}_F\left(\widehat{G}\right)\right]$ cannot reach to ${\varphi }_{\max }$ ($N=800,n=500$).

Figure 6

The difference of the average opinion of a threshold group by time $\left[\mathrm{\Delta }{\rho }_G(t,t+1)\right]$ for $\beta =-1$ and 1 ($N=800,n=500$). Negative coupling ($\beta =-1$) shows a sequential mechanism, and positive coupling ($\beta =1$) displays a segregated mechanism.