Conditional random matrix ensembles and the stability of dynamical systems
File(s)
Published version
Author(s)
Kirk, P
Rolando, DMY
MacLean, AL
Stumpf, MPH
Type
Journal Article
Abstract
Random matrix theory (RMT) has found applications throughout physics and applied mathematics, in subject areas as diverse as communications networks, population dynamics, neuroscience, and models of the banking system. Many of these analyses exploit elegant analytical results, particularly the circular law and its extensions. In order to apply these results, assumptions must be made about the distribution of matrix elements. Here we demonstrate that the choice of matrix distribution is crucial. In particular, adopting an unrealistic matrix distribution for the sake of analytical tractability is liable to lead to misleading conclusions. We focus on the application of RMT to the long-standing, and at times fractious, 'diversity-stability debate', which is concerned with establishing whether large complex systems are likely to be stable. Early work (and subsequent elaborations) brought RMT to bear on the debate by modelling the entries of a system's Jacobian matrix as independent and identically distributed (i.i.d.) random variables. These analyses were successful in yielding general results that were not tied to any specific system, but relied upon a restrictive i.i.d. assumption. Other studies took an opposing approach, seeking to elucidate general principles of stability through the analysis of specific systems. Here we develop a statistical framework that reconciles these two contrasting approaches. We use a range of illustrative dynamical systems examples to demonstrate that: (i) stability probability cannot be summarily deduced from any single property of the system (e.g. its diversity); and (ii) our assessment of stability depends on adequately capturing the details of the systems analysed. Failing to condition on the structure of dynamical systems will skew our analysis and can, even for very small systems, result in an unnecessarily pessimistic diagnosis of their stability.
Date Issued
2015-08-13
Date Acceptance
2015-07-01
Citation
New Journal of Physics, 2015, 17
ISSN
1367-2630
Publisher
IOP PUBLISHING LTD
Journal / Book Title
New Journal of Physics
Volume
17
Copyright Statement
© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
License URL
Sponsor
The Royal Society
Human Frontier Science Program
Biotechnology and Biological Sciences Research Council (BBSRC)
Grant Number
WRMA09R3/HLL
RGP0061/2011
BB/L023776/1
Subjects
Science & Technology
Physical Sciences
Physics, Multidisciplinary
Physics
stability
dynamical systems
control
MODEL-ECOSYSTEMS
COMPLEXITY
DIVERSITY
NETWORKS
BIODIVERSITY
Publication Status
Published
Article Number
083025