Robustness and fragility in coupled oscillator networks under targeted attacks

The dynamical tolerance of coupled oscillator networks against local failures is studied. As the fraction of failed oscillator nodes gradually increases, the mean oscillation amplitude in the entire network decreases and then suddenly vanishes at a critical fraction as a phase transition. This critical fraction, widely used as a measure of the network robustness, was analytically derived for random failures but not for targeted attacks so far. Here we derive the general formula for the critical fraction, which can be applied to both random failures and targeted attacks. We consider the effects of targeting oscillator nodes based on their degrees. First we deal with coupled identical oscillators with homogeneous edge weights. Then our theory is applied to networks with heterogeneous edge weights and to those with nonidentical oscillators. The analytical results are validated by numerical experiments. Our results reveal the key factors governing the robustness and fragility of oscillator networks.

Figure 1

The order parameter $\left|Z\right|$ vs. the inactivation fraction $p$ in the uniformly coupled model in Eq. (13) for a Barabási-Albert scale-free network with $N=100,\langle k\rangle /N=0.2$, $K=10$, $\mathrm{\Omega }=0.1$, and $a=b=1$. The cases of high-degree attack, low-degree attack, and random failure are indicated with downward triangles, upward triangles, and crosses, respectively.

Figure 2

The critical fraction $p_c$ vs. the coupling strength $K$ in the uniformly coupled model in Eq. (13) with $N=1000$ and $\mathrm{\Omega }=0.1$. The symbols indicate the mean of the numerical values of $p_c$ over 10 trials with different realizations of network structure and different initial conditions. The blue downward triangles, the red upward triangles, and the black crosses correspond to the high-degree attack (high), the low-degree attack (low), and the random failure (random), respectively. The error bars indicate the standard deviations. The lines indicate the mean of the theoretical values of $p_c$ obtained from Eqs. (14) and (16). The dash-dot, dashed, and solid lines correspond to the high-degree attack (high), the low-degree attack (low), and the random failure (random), respectively. (a) $a=b=0.5$, and (b) $a=0.05$ and $b=0.5$.

Figure 3

(a) The theoretical value of the critical fraction $p_c$ is plotted against the intrinsic parameter $b$ of the inactive oscillators. The order of $p_c$ for the three inactivation types is reversed at $b$ around 0.5. The other parameter values are the same as described in the caption of Fig. 2, but for $a=0.2$ and $K=10$. (b) The parameter conditions of $a$ and $b$, at which the reverse phenomenon occurs, are plotted for $K=5$, 10, and 15.

Figure 4

The critical fraction $p_c$ vs. the coupling strength $K$ in the weighted coupling model in Eq. (18) where $N=1000$, $\mathrm{\Omega }=0.1$, and $a=b=0.5$. The symbols and lines are the same as described in the caption of Fig. 2. The lines (theory) correspond to Eq. (20), but Eq. (22) also produces the similar results.

Figure 5

The critical fraction $p_c$ vs. the coupling strength $K$ in model in Eq. (25), where $N=1000$, $a=b=0.5$, and ${\mathrm{\Omega }}_j$ is randomly chosen from the uniform distribution on $[0,1]$. The symbols and lines are the same as described in the caption of Fig. 2. The lines (theory) are produced from Eq. (26).