Spectral theory for the stability of dynamical systems on large oriented locally tree-like graphs

The linear stability of large complex systems around their stationary points can be studied with random matrix theory. In one common model, the i.i.d. random matrix model, one assumes a simple statistical form for the interactions between distinct elements of a network. This model has been usefully applied in areas ranging from neuroscience and ecology to economics. One drawback, however, is that this model necessarily represents interactions on a dense graph – the average number of non-zero elements in each row or column of the interaction matrix diverges as a function of the number of elements n. This implies that any dynamical system should be unstable for n large enough, as the leading eigenvalue controlling the stability diverges as a function of n. This is plainly unrealistic, since many real systems of high dimension are apparently stable.
In a new paper, Izaak Neri and LML External Fellow Fernando Metz explore a more realistic scenario in which the system dynamics is defined by sparse random matrices. Such matrices reflect the non-random structures observed in real-world systems, such as networks with a prescribed degree distribution or with recurrent motifs. Due to the small number of interactions per element, dynamical systems associated
with sparse random matrices can be stable even for large values of n, making this model more acceptable for real-world networks. Neri and Metz develop exact mathematical methods to study the stability of large dynamical systems defined on sparse random graphs, showing that the eigenvalue distribution is not universal. However, the leading eigenvalue, as well as the statistics of the components of its associated right and left eigenvectors, do exhibit universal properties, which implies on universal behaviour of dynamical systems close to their stationary points.
In the paper, the researchers derive explicit analytical expressions for the leading eigenvalue as well as the first moment of its associated right and left eigenvectors. They specifically treat the case of oriented random graphs with prescribed degree distributions, allowing for correlations between in-degrees and out-degrees. Finally, they illustrate the theory using extensive numerical experiments, and apply the theory to the linear stability of large dynamical systems described by a set of randomly coupled differential equations, showing that the stability is governed by a few parameters characterizing the network structure. Their results help to clarify which network properties tend to stabilise or destabilise the stationary points in these systems, and should find practical applications in many fields of network science.
The paper is available at https://arxiv.org/abs/1908.07092

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