Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

TL;DR

The work addresses the stability of large heterogeneous linear systems by analyzing a fully connected random matrix with diagonal disorder. Using the cavity method and Hermitization, it derives the empirical spectral distribution and the spectrum boundary in the $n\to\infty$ limit, then links these spectral properties to fixed-point stability. A key finding is that antagonistic interactions ($\tau<0$) can stabilise fixed points for moderate interaction strength $\sigma$, whereas uncorrelated interactions ($\tau=0$) are destabilising, with explicit outlier formulas for $\mu\neq0$ and phase diagrams illustrating stability regions. These results extend May-type stability insights to heterogeneous diagonals and have implications for ecological and neural systems, as well as for future work on sparse networks and nonlinear dynamics.

Abstract

We analyse the stability of large, linear dynamical systems of variables that interact through a fully connected random matrix and have inhomogeneous growth rates. We show that in the absence of correlations between the coupling strengths, a system with interactions is always less stable than a system without interactions. Contrarily to the uncorrelated case, interactions that are antagonistic, i.e., characterised by negative correlations, can stabilise linear dynamical systems. In particular, when the strength of the interactions is not too strong, systems with antagonistic interactions are more stable than systems without interactions. These results are obtained with an exact theory for the spectral properties of fully connected random matrices with diagonal disorder.

Figures (7)

• Figure 1: Spectra of three random matrices $\mathbf{A}$ as defined in Eq. (\ref{['eq:AD']}) for the uncorrelated case $\tau=0$ and with diagonal elements that are independently drawn from a uniform distribution [Panel (a) and Panel(b)] or a Gaussian distribution [Panel (c)]. Markers denote the eigenvalues of a random matrix of size $n=3000$ and with off-diagonal elements $A_{ij} = J_{ij} + \mu/n$, where the $J_{ij}$ are drawn independently from a Gaussian distribution with zero mean and unit variance and where $\mu$ is as given in the subfigure captions. The red solid line denotes the solution to Eq. (\ref{['eq:OrD']}), which provides boundary of the support set $\mathcal{S}$ in the limit of infinitely large $n$. The eigenvalue outlier is indicated by an arrow in Panel (b). Panel (a) and (b) show the analytical solution Eq. (\ref{['eq:devCircular']}) and Panel (c) is obtained by numerically solving Eq. (\ref{['eq:OrD']}).
• Figure 2: Comparison between the spectra of two random matrices $\mathbf{A}$ with two different values of $\tau$. Eigenvalues plotted are for two matrices of size $n=3000$ whose diagonal elements are drawn from the bimodal distribution Eq. (\ref{['eq:bimodalx']}) with $d_-=-1$, $p=0.9$ and $d_+=0.1$, and whose off-diagonal entries are drawn from a normal distribution with zero mean $\mu=0$, variance $\sigma^2/n = 1/n$, and $\tau=0$ [Panel (a)] or $\tau=-0.7$ [Panel (b)]. The red line denotes the solution to the Eqs. (\ref{['re_g12']}) and (\ref{['eq:boundary_final']}).
• Figure 3: Comparison between the spectra of random matrices $\mathbf{A}$ with different values of the interaction strength $\sigma$. Eigenvalues plotted are for three matrices of size $n=3000$ whose off-diagonal elements $(J_{ij},J_{ji})$ are drawn from a joint Gaussian distribution with zero mean, a Pearson correlation coefficient $\tau=-0.7$, and a variance $\sigma^2/n$ as indicated. The diagonal elements follow a bimodal distribution with parameters $p=0.9$, $d_-=-1,d_+=0.1$, and $\mu=0$. The red line denotes the solution to the Eqs. (\ref{['re_g12']}) and (\ref{['eq:boundary_final']}).
• Figure 4: Effect of the interaction strength $\sigma$ on the real part of the leading eigenvalue $\lambda_1$ when $\mu=0$ and for $\tau = 0$ (triangle, dotted), $\tau=-0.8$ (circle, solid) and $\tau=-1$ (diamond, dashed). Lines show the Eq. (\ref{['eq:lambda1Analyt2']}). Markers are numerical results obtained for random matrices $\mathbf{A}$ with diagonal elements $D_j$ that are drawn independently from a uniform $p_D$ supported on the interval $[d_-,d_+] = [-1,0.1]$ and with pairs of off-diagonal elements $(J_{ij},J_{ji})$ that are drawn independently from a normal distribution with mean $0$, variance $\sigma^2/n$, and Pearson correlation coefficient $\tau$ as provided. Each marker represents the largest eigenvalue of one matrix realisation of size $n=7000$.
• Figure 5: Effect of the interaction strength $\sigma$ on the real part of the leading eigenvalue $\lambda_1$ for random matrices with $\mu=2$ and all other parameters the same as in Fig. \ref{['fig:reEntranceEffect']}. Similar to Fig. \ref{['fig:reEntranceEffect']}, solid lines/circles correspond with $\tau=-0.8$ and dashed lines/triangles with $\tau=-1$. Gray lines show Eq. (\ref{['eq:lambda1Analyt2']}). Black lines show the maximum between Eq. (\ref{['outlier_uniform']}), for the eigenvalue outlier, and Eq. (\ref{['eq:lambda1Analyt2']}), for the leading eigenvalue of $\mathcal{S}$. Each marker represents the largest eigenvalue of one matrix realisation of size $n=3000$.