FIG. 3: Representative trajectories of the macroscopic vari- able S ( t ). Note the large amplitude of the oscillations in the intermediate case p = 0 . 44. The “signal” is the periodic func- tion A sin(2 πt/T ). Values of N , C and T as in Fig. 2, A = 0 . 2. FIG. 2: Spectral amplification factor R versus probability of repulsive links p . Inset: the influence of the amplitude A of the external forcing on the response R . For suprathreshold amplitudes, A ≳ 0 . 4, R decreases with A due to the denomi- nator A 2 in the definition of the spectral amplification factor R . T = 300, N and C as in Fig. 1. 3 0 500 1000 1500 2000 2500 3000 3500 t -1 -0.5 0 0.5 1 S p=0.0 p=0.44 p=1.0 Signal 0.3 0.4 0.5 0.6 p 0 5 10 15 20 R A=0.1 A=0.2 A=0.3 A=0.4 A=0.5 0 0.2 0.4 0.6 0.8 1 A 0 5 10 15 20 R p=0.50 p=0.44 p=0.43 p=0.0 0.3 0.4 0.5 0.6 p 0 5 10 15 20 25 30 R T=50 T=100 T=300 T=500 0 200 400 600 800 1000 T 0 5 10 15 20 25 R p=0.44 We now turn on the forcing A > 0 and study the sys- tem response, as measured by the spectral power ampli- fication factor R defined above, Eq. (3). Consider first the case p = 0. For a small, sub-threshold, amplitude A the macroscopic variable S ( t ) will just execute small oscillations of amplitude proportional to A around the stable values S = +1 or S = − 1. As A increases beyond the threshold value A$_{o}$ ≈ 0 . 4 the amplitude of the forcing is large enough to induce large jumps of the macroscopic variable from S ≈ − 1 to S ≈ +1 and vice versa. This change of behavior at A$_{o}$ appears as a sudden increase in the value of R , as shown in the inset of Fig. 2. As the same inset shows, similar behavior is observed for 0 < p ≲ p$_{c}$ : the response shows a sudden increase for a particular value of the amplitude A and then decreases monotonically. For p > p$_{c}$ , the response is very small and almost independent on the value of A . More interesting, and the main result of this paper, is the dependence of R on the probability p of repulsive links, main plot in Fig. 2. We note that there is an opti- mal probability of repulsive links that is able to amplify signals whose amplitude would be sub-threshold in the case p = 0, i.e. A < A$_{o}$ . For suprathreshold signals, A > A$_{o}$ , the presence of repulsive links does no longer lead to enhanced amplification. As shown in the figure, the optimal value for amplification is close to the criti- cal value p$_{c}$ signaling the transition from bistability to monostability in the non-forced case. The optimal am- plification as a function of p can clearly be observed in Fig.(3) which shows representative trajectories for p = 0 (small oscillations around the value S = +1), p = p$_{c}$ (large oscillations between S ≈ +1 and S ≈ − 1) and p = 1 (small oscillations around S = 0). The existence of an optimal value of the fraction of repulsive p for which signal amplification is maximum is somehow reminiscent of the stochastic resonance phe- nomenon. There are some important differences, how- ever. While in stochastic resonance, the response R shows a maximum as a function of period, resulting from the matching between Kramers’ rate and the forcing half- period, in our case the same optimal disorder p amplifies responses to signals of every period, as shown in Fig. 4. When the signal is slow enough, the system has time to respond to the fuller extent, going to the absolute extrema of the potential, and the amplification factor reaches a constant value, see inset of Fig. 4. It is possible to reinterpret these results in terms of the effective potential V eff ( S ) introduced above. The pe- FIG. 4: The influence of the signal period T on the response R . In the inset, we see the response R reaches a constant value for slow enough signals. Values of N , C and A as in Fig. 3.