Random antagonistic matrices

@article{Cicuta2016RandomAM,
  title={Random antagonistic matrices},
  author={Giovanni M Cicuta and Luca Guido Molinari},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2016},
  volume={49}
}
The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the entries  i , j and  j , i are real and have opposite signs, or are both zero, and the diagonal is zero. This generalization of antisymmetric matrices is suggested by the linearized dynamics of competitive species in ecology. 

Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

We analyse the stability of large, linear dynamical systems of variables that in-teract through a fully connected random matrix and have inhomogeneous growth rates. We show that in the absence of

Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations.

An exact theory is developed for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices of large dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions that reveals that the nature of local interactions have a strong influence on system's stability.

References

SHOWING 1-10 OF 39 REFERENCES

The stability of a feasible random ecosystem

The weight of the evidence, and the beliefs of most biologists, seem to support the view that ecosystems tend to be more stable, the larger the number of interacting species they contain, and that complexity makes for instability.

Scaling properties of band random matrices.

It is shown on the basis of numerical data that the normalized localization length of eigenvectors of band random matrices follows a scaling law. The scaling parameter is b 2 /N, where b measures the

Eigenvalue spectra of random matrices for neural networks.

Eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances are computed.

Scaling properties of the eigenvalue spacing distribution for band random matrices

The authors show that the spacing distribution for eigenvalues of band random matrices is described by a single parameter b2/N, where b is the band half-width and N is the size of the matrices. It is

On the density of states of sparse random matrices

The supersymmetric method of calculation of the density of states of a sparse random matrix is shown to be absolutely equivalent to the replica trick. A functional generalization of the

Elliptic law for real random matrices

In this paper we consider ensemble of random matrices $\X_n$ with independent identically distributed vectors $(X_{ij}, X_{ji})_{i \neq j}$ of entries. Under assumption of finite fourth moment of

Conditional random matrix ensembles and the stability of dynamical systems

A statistical framework is developed that reconciles the two contrasting approaches of RMT to the long-standing, and at times fractious, ‘diversity-stability debate’, concerned with establishing whether large complex systems are likely to be stable.

The stability–complexity relationship at age 40: a random matrix perspective

Since the work of Robert May in 1972, the local asymptotic stability of large ecological systems has been a focus of theoretical ecology. Here we review May’s work in the light of random matrix

Universality and the circular law for sparse random matrices.

This paper proves a universality result for sparse random n by n matrices where each entry is nonzero with probability $1/n^{1-\alpha}$ where $0<\alpha\le1$ is any constant.

Localization in non-Hermitian chains with excitatory/inhibitory connections

We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of