Stability of large complex systems with heterogeneous relaxation dynamics

Abstract

We study the probability of stability of a large complex system of size N within the framework of a generalized May model, which assumes a linear dynamics of each population size n i (with respect to its equilibrium value): d n i d t = − a i n i − T ∑ j J i j n j . The a i > 0’s are the intrinsic decay rates, J ij is a real symmetric (N × N) Gaussian random matrix and T measures the strength of pairwise interaction between different species. Our goal is to study how inhomogeneities in the intrinsic damping rates a i affect the stability of this dynamical system. As the interaction strength T increases, the system undergoes a phase transition from a stable phase to an unstable phase at a critical value T = T c. We reinterpret the probability of stability in terms of the hitting time of the level b = 0 of an associated Dyson Brownian motion (DBM), starting at the initial position a i and evolving in ‘time’ T. In the large N → ∞ limit, using this DBM picture, we are able to completely characterize T c for arbitrary density μ(a) of the a i ’s. For a specific flat configuration a i = 1 + σ i − 1 N , we obtain an explicit parametric solution for the limiting (as N → ∞) spectral density for arbitrary T and σ. For finite but large N, we also compute the large deviation properties of the probability of stability on the stable side T < T c using a Coulomb gas representation.

Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

“…Expanding the Eqs. (17) in small values of g 11 > 0 and g 22 > 0, we obtain that for all values of z ∈ S it holds that…”

“…Notably, for the symmetric case with τ = 1 Pastur derived a functional equation that determines ρ [15]. Recently, the symmetric case was revisited in [17], and in that reference also the large deviations of λ 1 were computed in the case when the matrix entries J ij are drawn from a Gaussian distribution; note that large deviations are not universal and depend on the statistics of (J ij , J ji ) as determined by the distribution p J 1 ,J 2 . In the case when τ = 0 and p D is a bimodal distribution the spectral distribution ρ has been determined in Refs.…”

“…In this paper, we relax this condition and consider random matrix models with growth rates A jj = D j that fluctuate from one variable to the other. In the symmetric case (τ = 1), such random matrices are called deformed Wigner matrices [15][16][17] and in this case a functional equation that determines the spectral distribution in the limit of large n has been derived by Pastur in Ref. [15].…”

“…( 8) for which g 11 > 0 and g 22 > 0 and ∂ z * g 21 = 0, yielding the probability distribution ρ > 0 for z ∈ S Although the trivial solution solves the set of Eqs. (17) for any value of z and for η = 0, it is only for z / ∈ S that the trivial solution is relevant. Indeed, when z ∈ S the trivial solution is unstable with respect to infinitesimal small perturbations, and hence the regulator η > 0 guarantees that the spectral distribution for z ∈ S is determined by the nontrivial solution.…”

“…As a consequence, the boundary of the support set S follows from a linear stability analysis of the Eqs. (17) around the trivial solution [7]. Expanding the Eqs.…”