Dynamics of rumor spreading in complex networks

We derive the mean-field equations characterizing the dynamics of a rumor process that takes place on top of complex heterogeneous networks. These equations are solved numerically by means of a stochastic approach. First, we present analytical and Monte Carlo calculations for homogeneous networks and compare the results with those obtained by the numerical method. Then, we study the spreading process in detail for random scale-free networks. The time profiles for several quantities are numerically computed, which allows us to distinguish among different variants of rumor spreading algorithms. Our conclusions are directed to possible applications in replicated database maintenance, peer-to-peer communication networks, and social spreading phenomena.

Figure 1

Time evolution of the density of stifler individuals for different values of $\alpha$. From below, the values of $\alpha$ go from $1.0$ to $0.1$ at fixed increments of $0.1$. The inset shows the time dependence of the density of spreaders. The system size is $N={10}^4$, $⟨k⟩=6$, and $\gamma =3$. Time is in units of ${\alpha }^{-1}$.

Figure 2

Density of ignorants $i_k$ at the end of the rumor process as a function of their connectivity $k$. A clear exponential decay can be appreciated for all values of $\alpha$ shown. This implies that hubs have efficiently learned the rumor.

Figure 3

Density of stiflers as a function of time for $\alpha =0.1$ (main figure) and $\alpha =1.0$ (inset) when the initial spreader has the connectivity indicated in the inset. Note that in all cases the final density of individuals who have learned the rumor is the same, but the asymptotic value is reached at different times. The model parameters are as in Fig. 1. Time is in units of ${\alpha }^{-1}$.

Figure 4

Stifler’s growth as a function of time for three different annihilation mechanisms as explained in the text. Curves show the time profiles for the two extreme values of $\alpha$ used in the simulations: $\alpha =1.0$ (lower curves) and $0.1$ (higher curves). The curve for $s\text{−}s$ interactions is for $\alpha =0.1$ and is not complete for clarity. Time is in units of ${\alpha }^{-1}$.

Figure 5

Growth and decay of the populations of spreaders when the annihilation mechanism includes interactions of the type $s\text{−}s$ and $s\text{−}r$ or only $s\text{−}r$ or $s\text{−}s$. Curves show the time profiles for the two extreme values of $\alpha$ used in the simulations: $\alpha =1.0$ (lower curves) and $0.1$ (higher curves). Note that although the final number of stiflers when only $s\text{−}s$ interactions enter in the decay mechanism is low, the time it takes for the rumor to reach the asymptotic value is very high as compared to the other two mechanisms. Curves corresponding to the $s\text{−}s$ interactions are not complete for clarity. Time is in units of ${\alpha }^{-1}$.