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of a larger system is best amplified at a higher probability of repulsive links. As the fraction of repulsive links must not exceed the fraction of positive ones, there can be a limit on how large can a system be, to be able to amplify a signal. The same behavior, focusing on the number of neighbors was found in a previous study of an Ising-like network model [28]. 

We have already commented that the optimal prob- ability of repulsive links drives the system to a glassy 

## IV. SPECTRAL ANALYSIS 

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phase. Anderson [34, 35] has proposed a connection be- tween a glass and a delocalization-localization transition, relating the existence of many metastable states with a localization of modes. From this proposal, we retain the idea to work in the eigenspace of the interaction matrix, and to look for the fraction of repulsive links where mode localization becomes significant. This approach has the virtue of not only identifying the steady states, but also to shed light onto how the reaction to perturbations is sustained and spreads along the system, depending on the fraction of repulsive links. In this manner, we hope to locate the region where multistability is expected, and also to understand the mechanism of response to external p erturbations. 

Following [27], let us define the eigenvalues Q$_{α}$ and (normalized) eigenvectors e α = ( e α $_{1}$, . . . , e α $_{N}$) of the Lapla- cian matrix [29] J ′ ij : 

The effect of the competitive interactions can be de- scribed by the so-called participation ratio of eigenvec- tor e $^{α}$, defined as PR$_{α}$ = 1 / ∑ N i $_{=1}$[ e α i ] $^{4}$. It quantifies the number of components that participate significantly in each eigenvector. A state α with equal components has PR$_{α}$ = N , and one with only one component has PR$_{α}$ = 1. When PR$_{α}$ = 1 on a fraction f of elements, and 0 elsewhere, then PR$_{α}$ = f , which justifies its name. More precisely, we will define “localized” modes as the ones whose participation ratio is less than 0 . 1 N . Our first observation (Fig. 10) is that at the optimal region 

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