Drastic disorder-induced reduction of signal amplification in scale-free networks

Understanding information transmission across a network is a fundamental task for controlling and manipulating both biological and manmade information-processing systems. Here we show how topological resonant-like amplification effects in scale-free networks of signaling devices are drastically reduced when phase disorder in the external signals is considered. This is demonstrated theoretically by means of a starlike network of overdamped bistable systems, and confirmed numerically by simulations of scale-free networks of such systems. The taming effect of the phase disorder is found to be sensitive to the amplification's strength, while the topology-induced amplification mechanism is robust against this kind of quenched disorder in the sense that it does not significantly change the values of the coupling strength where amplification is maximum in its absence.

Figure 1

Theoretical average amplification $〈G_{\mathrm{eff}}〉$ in the $\left(k-\lambda \right)$ parameter plane with $\lambda \in \left[0,0.0035\right]$ and $k\in \left[0,1\right]$ [left panel, Eq. (9)] and corresponding numerical results $〈〈G〉〉$ (right panel) for a starlike network [see Eq. (3)] and $N=500,\omega =2\pi \times {10}^{-1},\tau =0.01$.

Figure 2

Top panel: Average amplification $〈〈G〉〉$ versus coupling $\lambda$ for a BA scale-free network and four values of the phase disorder parameter: $k=0$ (black $+)$, $k=0.3$ (red $\times )$, $k=0.5$ (blue stars), and $k=0.7$ (green squares). Note that the first relative maximum of $〈〈G〉〉$ occurs around $\lambda \approx 0.008$ for the four values of $k$ and the network has a maximal active hub having 136 leaves. For this effective starlike network the theoretically predicted maximum occurs at $\lambda ={\lambda }_{max,1}\approx 0.0074$. Bottom panel: Average amplification $〈〈G〉〉$ versus phase disorder parameter $k$ for a BA scale-free network and three values of the coupling: $\lambda =0.009$ (blue +), $\lambda =0.015$ (black $\times )$, and $\lambda =0.045$ (red squares). Averaged degree $〈\kappa 〉=3,\gamma =2.7$, and the remaining fixed parameters are as in Fig. 1.

Figure 3

Average of the average amplification over ${10}^2$ random realizations of the network connectivity $〈〈〈G〉〉〉\equiv 〈{max}_i{x}_i/\tau 〉$ for two values of the phase disorder parameter $\mathrm{[}k=0$(black circles) and $k=0.7$(red squares)] (top panel) and corresponding synchronization coefficient $〈\rho 〉$ [see Eq. (2)] for $k=0$ and $k=0.7$ (bottom panels) versus coupling $\lambda$ for a BA scale-free network with $\gamma =2.7$ and $〈\kappa 〉=3$. Other fixed parameters are as in Fig. 1.

Figure 4

Average of the average amplification over 30 random realizations of the network connectivity $〈〈〈G〉〉〉\equiv 〈{max}_i{x}_i/\tau 〉$ (top panel) and corresponding synchronization coefficient $〈\rho 〉$ [see Eq. (2), medium and bottom panels] versus coupling $\lambda$ for a BA scale-free network with $〈\kappa 〉=3$ and different values of the phase disorder parameter and the power-law distribution exponent: $(k,\gamma )=(0.1,2)$ (black +), $\left(0.1,2.5\right)$ (red $\times )$, $(0.5,2)$ (green squares), $\left(0.5,2.5\right)$ (blue circles). Other fixed parameters are as in Fig. 1.

Figure 5

Average of the average amplification over 30 random realizations of the network connectivity $〈〈〈G〉〉〉\equiv 〈{max}_i{x}_i/\tau 〉$ (top panel) and corresponding synchronization coefficient $〈\rho 〉$ [see Eq. (2), bottom panel] versus coupling $\lambda$ for a BA scale-free network with $〈\kappa 〉=5,\gamma =2.7$, and three values of the phase disorder parameter: $k=0$ (black circles), $0.1$ (red triangles), $0.5$ (green squares). Other fixed parameters are as in Fig. 1.

Figure 6

Average amplification $〈〈G〉〉$ versus coupling $\lambda$ for a BA scale-free network with $N=2\times {10}^3,\omega =2\pi \times {10}^{-1},\tau =0.01$, and two values of the phase disorder parameter: $\kappa =0$ (black +) and $\kappa =0.7$ (red squares). The two first maxima appear around the expected $\lambda$ values according to the degrees of the main hubs of this network (1152 and 687, respectively).