Network modularity controls the speed of information diffusion

The rapid diffusion of information and the adoption of social behaviors are of critical importance in situations as diverse as collective actions, pandemic prevention, or advertising and marketing. Although the dynamics of large cascades have been extensively studied in various contexts, few have systematically examined the impact of network topology on the efficiency of information diffusion. Here, by employing the linear threshold model on networks with communities, we demonstrate that a prominent network feature—the modular structure—strongly affects the speed of information diffusion in complex contagion. Our simulations show that there always exists an optimal network modularity for the most efficient spreading process. Beyond this critical value, either a stronger or a weaker modular structure actually hinders the diffusion speed. These results are confirmed by an analytical approximation. We further demonstrate that the optimal modularity varies with both the seed size and the target cascade size and is ultimately dependent on the network under investigation. We underscore the importance of our findings in applications from marketing to epidemiology, from neuroscience to engineering, where the understanding of the structural design of complex systems focuses on the efficiency of information propagation.

Figure 1

Simulation results (dots) and analytical predictions (lines) of Eq. (5) (cf. Materials and Methods). Blue axis: the average speed of information diffusion, $\overline{v}$. Red axis: the size of information cascade, ${\rho }_{t_s}$. The $x$ axis represents the strength of network modularity controlled by $\mu$. Green area: the range of $\mu$ that can enable global cascades (${\rho }_{t_s}⩾0.99$). The dashed vertical line corresponds to $\mu =0.17$ that yields the highest $\overline{v}$ by prediction. The simulation results are averaged over 100 realizations of the network for each $\mu$, with $N=1\times {10}^5,z=10,{\rho }_0=0.1,\theta =0.35,f=0.01$. The error bars indicate the interquartile ranges.

Figure 2

Cross sections of three different $\mu$ values in Fig. 1 that enable global cascades. (a, d) $\mu =0.13$; (b, e) $\mu =0.17$; (c, f) $\mu =0.21$. (a–c) The diffusion speed $v_t^{\left(i\right)}$ in $C_1$ and $C_2$ as a function of time step $t$. (d–f) Same as (a)–(c), but for the cumulative cascade size ${\rho }_t^{\left(i\right)}$. The theoretical predictions of Eq. (4) (lines) show excellent agreement with the numerical simulations (dots), averaged over 100 runs. The optimal $\mu =0.17$ achieves the shortest total diffusion time, thus the highest average diffusion speed.

Figure 3

Phase diagrams of the average diffusion speed $\overline{v}$ as a function of seed size ${\rho }_0$ (a) and threshold $\theta$ (b) on SBM networks. The two red curves mark the region for global cascades. The blue curve represents the $\mu$ value that yields the highest $\overline{v}$ for a given ${\rho }_0$ or $\theta$. The results are based on simulations, averaged over 100 runs for each combination of $({\rho }_0,\mu )$ or $(\theta ,\mu )$. Simulation parameters are $N=1\times {10}^5,z=10,f=0.01$, with $\theta =0.35$ (a) and ${\rho }_0=0.1$ (b). The seeds are randomly selected from a single community. The dashed line is a slice at ${\rho }_0=0.1$ ($\theta =0.35$) in Fig. 1.

Figure 4

Phase diagrams of the average diffusion speed $\overline{v}$ on the LFR (a) and Twitter (b) networks. Parameters $\mu$ and $p$ on the $x$ axis control the network modularity. The blue curve indicates the optimal $\mu$ (or $p$) for $\overline{v}$ for a given seed size ${\rho }_0$. The normalized modularity $Q_{\text{norm}}$ with respect to $\mu$ (or $p$) is shown on the top axis. Network statistics are $N=25000,z=10,\gamma =2.5,\beta =1.5,k_{\max }=30$ (for LFR); $N=81306,z=16$ (for Twitter). Simulations are averaged over 100 runs, with $\theta =0.3,f=0.01$. The seeds are randomly selected across the whole network.

Figure 5

The optimal network modularity for fast information diffusion changes as a function of the target cascade size on three different types of networks. The modularity is controlled by $\mu$ for SBM (a) and LFR (b) networks [or by $p$ for the Twitter network (c)]. The optimal values of $\mu$ and $p$ are selected from those that can achieve the given target size $\mathrm{\Omega }$. All simulation parameters are the same as in Fig. 3 and Fig. 4. Note that a small ${\rho }_0$ may not be able to reach all target cascade sizes, e.g., a seed size of ${\rho }_0=0.1$ on the Twitter network is able to infect only up to about $\mathrm{\Omega }=65\%$ of all nodes. Line plots for different ${\rho }_0$ start from different $\mathrm{\Omega }$ since $\mathrm{\Omega }>{\rho }_0$.