Thesis title: Sparse non-Hermitian random graphs: spectral features and applications in theoretical ecology
In this thesis, we investigate the spectral properties of sparse non-Hermitian random matrices, with a specific interest on their implications for ecological models. The results discussed in this thesis reveal that considering sparse and asymmetric interactions opens the door to a plethora of new spectral results which can provide valuable insights into the modelling of complex ecosystems.
In general, the spectra of sparse random graphs are characterised by spectral tails accumulating on the real axis and elongating with the system size. Concise mathematical results for these tails have proved elusive, especially in case of asymmetric matrices. Exploiting the fact that these eigenvalues are associated to localised eigenvectors, we derive simple analytical expressions describing the asymptotic behaviour of the spectral density in the tails region.
Moreover, we identify a property, which we named strong local sign stability, characterising an exceptional class of graphs for which spectral tails are either absent or accumulate exclusively along the imaginary axis, leading to significant implications for the dynamics of the associated models. At the ecological level, these findings indicate that moving beyond traditional, fully connected models is a viable strategy for overcoming paradoxical implications and achieving a clearer understanding of the intricate balance within real-world ecological networks.
Additionally, the spectra of certain strongly locally sign stable sparse random graphs with low connectivity exhibit distinctive reentrances near the real axis, resulting in leading eigenvalues with non-zero imaginary part. This effect persists also in presence of disordered diagonal and stripy structure, provided the disorder is not too strong and the connectivity is low enough. In this thesis, we identify various transitions regarding the leading eigenvalue and the presence of the reentrances, in terms of both connectivity and diagonal disorder.
More specifically, in Chapter 1 we provide a basic introduction to theoretical ecology, starting from models of individual ecological interactions to models of entire species-rich ecosystems, and showing how it is possible to assess various stability properties of these models by studying the spectral properties of related matrices.
In Chapter 2 we present some fundamental results in random matrix theory, starting from the classical results describing the spectral features of dense random matrices up to the most recent developments concerning sparse non-Hermitian matrices. We then identify the peculiar spectral features of the sparse matrices which we are going to delve into in the following chapters.
In Chapter 3 we investigate the dependence of the real part of leading eigenvalue of sparse random graphs on matrix size. We introduce strong local sign stability as a simple, general, criterion to predict whether the real part of the leading eigenvalue is indefinitely growing with the matrix size or not.
Then, in Chapter 4 we focus on the spectra of low connectivity Erdős-Rényi graphs with predator-prey interactions, which exhibit the distinctive reentrance effect. Through extensive numerical investigations we obtain a phase diagram describing this effect in terms of connectivity and diagonal disorder.
Finally, in Chapter 5 we provide a quantitative theory for the spectral tails of sparse non-Hermitian random matrices, describing both the spectral density and the inverse participation ratio, and we present numerical results matching our analytical predictions.