Impulse-induced optimum signal amplification in scale-free networks

Optimizing information transmission across a network is an essential task for controlling and manipulating generic information-processing systems. Here, we show how topological amplification effects in scale-free networks of signaling devices are optimally enhanced when the impulse transmitted by periodic external signals (time integral over two consecutive zeros) is maximum. This is demonstrated theoretically by means of a star-like network of overdamped bistable systems subjected to generic zero-mean periodic signals and confirmed numerically by simulations of scale-free networks of such systems. Our results show that the enhancer effect of increasing values of the signal's impulse is due to a correlative increase of the energy transmitted by the periodic signals, while it is found to be resonant-like with respect to the topology-induced amplification mechanism.

Figure 1

Signal $f\left(t;T,m\right)$ [Eq. (12)] vs $t/T$, where $T$ is the period and $N\left(m\right)\equiv 1/\left\{a+b/[1+exp(\left\{m-c\right\}/d)]\right\}$, with $a\equiv 0.43932, b\equiv 0.69796, c\equiv 0.3727$, and $d\equiv 0.26883$, for four values of the shape parameter: $m=0$ (sinusoidal pulse), $m=0.72\simeq m_{max}$ (nearly square-wave pulse), $m=0.99$ (double-humped pulse), and $m=1-{10}^{-6}$ (sharp double-humped pulse). Inset: The normalized impulse $I(m,T)/I(0,T)$ [Eq. (13)] vs $m$.

Figure 2

Normalized average amplification $〈G\left(m\right)〉/〈G(m=0)〉$ vs shape parameter $m$ for an isolated overdamped bistable system [Eq. (1) with $\lambda =0]$ subjected to a signal given by Eq. (12) and seven values of the signal period: $T=1/\sqrt{2}$ (circles), $T=1$ (pluses), $T=\sqrt{2}$ (triangles), $T=2\pi /\sqrt{2}$ (stars), $T=7$ (diamonds), $T=10$ (squares), and $T=50$ (crosses). The solid line shows the normalized impulse $I(m,T)/I(0,T)$ [Eq. (13)] vs shape parameter $m$, while the dashed vertical line indicates the value $m=m_{max}\simeq 0.717$ for which $I(m,T)/I(0,T)$ presents its single maximum as a function of $m$. Notice that the period of the linearized motion around any of the potential minima, $T=2\pi /\sqrt{2}$, corresponds to the shortest significant time scale.

Figure 3

Theoretical average amplification $〈G〉$ in the $\left(\lambda -m\right)$ parameter plane [left panel; Eqs. (11) and (12)] and corresponding numerical results $〈G〉$ (right panel) for a star-like network [Eq. (6)] with $\lambda \in \left[0,0.01\right],m\in \left[0,0.99\right]$, and $N=500,T=10,\tau =0.01$. The dashed lines indicate the values $\lambda ={\lambda }_{max}\simeq 0.002, m=m_{max}\simeq 0.717$ for which $〈G〉$ presents its single maximum in the $\left(\lambda -m\right)$ parameter plane.

Figure 4

Top: $〈G〉$ vs $\lambda$ for a BA scale-free network, $T=10$ and four values of the shape parameter: $m=0$ (pluses), $m=0.72\simeq m_{max}$ (crosses), $m=0.99$ (stars), and $m=1-{10}^{-6}$ (squares). Note that the first relative maximum of $〈G〉$ occurs around $\lambda \approx 0.008$ for the four values of $m$ while the network has a maximal active hub with 136 leaves. For this effective star-like network the theoretically predicted maximum occurs at $\lambda ={\lambda }_{max,1}\approx 0.0074$. Middle: $〈G〉$ vs $m$ for a BA scale-free network, $T=10$, and three values of the coupling: $\lambda =0.009$ (pluses), $\lambda =0.015$ (crosses), and $\lambda =0.045$ (squares). Bottom: $〈G〉$ vs $m$ for $T=1,\lambda =0.015$ and two external signals: $f(t;T,m)$ [Eq. (12); stars] and $g(t;T,m)$ (see the text). The dashed vertical and horizontal lines indicate the values $m=m_{max}\simeq 0.717$ and $〈G〉\left(m=0\right)$, respectively. Averaged degree $〈\kappa 〉=3, \gamma =2.7, N=500, \tau =0.01$.

Figure 5

Average of the average amplification over ${10}^2$ random realizations of the network connectivity $\langle \langle G\rangle \rangle \equiv \langle {max}_i{x}_i/\tau \rangle$ for two values of the shape parameter $[m=0$ (circles) and $m=0.72\simeq m_{max}$ (squares); top panel] and corresponding synchronization coefficient $〈\rho 〉$ [Eq. (2)] for $m=0$ and $m=0.72$ (bottom panel) vs coupling $\lambda$ for a BA scale-free network with $〈\kappa 〉=3,\gamma =2.7$. Other fixed parameters are as in Fig. 3.

Figure 6

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 shape parameter $[m=0$ (triangles) and $m=0.717\simeq m_{max}$ (circles); top panel] and corresponding synchronization coefficient $〈\rho 〉$ (2) for $m=0$ and $m=0.717$ (bottom panel) vs coupling $\lambda$ for a BA scale-free network with $N=2\times {10}^3,〈\kappa 〉=3,\gamma =2.7$. Other fixed parameters are as in Fig. 3.