arXiv subjects

Pietro Valigi

Publications and source records attributed to Pietro Valigi.

5 recordsLinked to original sources

arXiv2026

Spectral properties and phase diagrams of sparse antagonistic random matrices with diagonal disorder and Jacobian-like structure

Complex interacting systems are often modelled by random matrices whose spectral properties dictate stability. In sparse antagonistic matrices without diagonal disorder, low connectivity gives rise to a characteristic reentrance effect in the spectral boundary near the real axis, which disappears via a continuous transition as the connectivity increases. The reentrance effect implies the presence of a complex leading eigenvalue, which suggests the existence of a phase characterized by oscillatory dynamics around equilibrium. Here, we expand the investigation to matrices featuring diagonal disorder and a Jacobian-like structure. In these settings, the spectrum also develops a segment of eigenvalues accumulating on the real axis, which can trigger a discontinuous jump of the complex leading eigenvalue to a purely real value. The interplay between connectivity and disorder produces a rich variety of spectral behaviours. Employing the cavity method and a an adaptation of the Population Dynamics algorithm, we map a phase diagram with five distinct spectral phases. Finally, we show that the algorithm underestimates the spectral support under strong disorder, motivating future technical developments to handle this limit.

cond-mat.dis-nn↗

arXiv2025

Graphicality of power-law and double power-law degree sequences

The graphicality problem -- whether or not a sequence of integers can be used to create a simple graph -- is a key question in network theory and combinatorics, with many important practical applications. In this work, we study the graphicality of degree sequences distributed as a power-law with a size-dependent cutoff and as a double power-law with a size-dependent crossover. We combine the application of exact sufficient conditions for graphicality with heuristic conditions for nongraphicality which allow us to elucidate the physical reasons why some sequences are not graphical. For single power-laws we recover the known phase-diagram, we highlight the subtle interplay of distinct mechanisms violating graphicality and we explain why the infinite-size limit behavior is in some cases very far from being observed for finite sequences. For double power-laws we derive the graphicality of infinite sequences for all possible values of the degree exponents $\gamma_1$ and $\gamma_2$, uncovering a rich phase-diagram and pointing out the existence of five qualitatively distinct ways graphicality can be violated. The validity of theoretical arguments is supported by extensive numerical analysis.

cond-mat.dis-nn↗

arXiv2025

Local equations for the generalized Lotka-Volterra model on sparse asymmetric graphs

Real ecosystems are characterized by sparse and asymmetric interactions, posing a major challenge to theoretical analysis. We introduce a new method to study the generalized Lotka-Volterra model with stochastic dynamics on sparse graphs. By deriving local Fokker-Planck equations and employing a mean-field closure, we can efficiently compute stationary states for both symmetric and asymmetric interactions. We validate our approach by comparing the results with the direct integration of the dynamical equations and by reproducing known results and, for the first time, we map the phase diagram for sparse asymmetric networks. Our framework provides a versatile tool for exploring stability in realistic ecological communities and can be generalized to applications in different contexts, such as economics and evolutionary game theory.

q-bio.PE↗

arXiv2025

Eigenvalue spectral tails and localization properties of asymmetric networks

In contrast to the neatly bounded spectra of densely populated large random matrices, sparse random matrices often exhibit unbounded eigenvalue tails on the real and imaginary axis, called Lifshitz tails. In the case of asymmetric matrices, concise mathematical results have proved elusive. In this work, we present an analytical approach to characterising these tails. We exploit the fact that eigenvalues in the tail region have corresponding eigenvectors that are exponentially localised on highly-connected hubs of the network associated to the random matrix. We approximate these eigenvectors using a series expansion in the inverse connectivity of the hub, where successive terms in the series take into account further sets of next-nearest neighbours. By considering the ensemble of such hubs, we are able to characterise the eigenvalue density and the extent of localisation in the tails of the spectrum in a general fashion. As such, we classify a number of different asymptotic behaviours in the Lifshitz tails, as well as the leading eigenvalue and the inverse participation ratio. We demonstrate how an interplay between matrix asymmetry, network structure, and the edge-weight distribution leads to the variety of observed behaviours.

cond-mat.dis-nn↗

arXiv2023

Local sign stability and its implications for spectra of sparse random graphs and stability of ecosystems

We study the spectral properties of sparse random graphs with different topologies and type of interactions, and their implications on the stability of complex systems, with particular attention to ecosystems. Specifically, we focus on the behaviour of the leading eigenvalue in different type of random matrices (including interaction matrices and Jacobian-like matrices), relevant for the assessment of different types of dynamical stability. By comparing the results on Erdos-Renyi and Husimi graphs with sign-antisymmetric interactions or mixed sign patterns, we introduce a sufficient criterion, called strong local sign stability, for stability not to be affected by system size, as traditionally implied by the complexity-stability trade-off in conventional models of random matrices. The criterion requires sign-antisymmetric or unidirectional interactions and a local structure of the graph such that the number of cycles of finite length do not increase with the system size. Note that the last requirement is stronger than the classical local tree-like condition, which we associate to the less stringent definition of local sign stability, also defined in the paper. In addition, for strong local sign stable graphs which show stability to linear perturbations irrespectively of system size, we observe that the leading eigenvalue can undergo a transition from being real to acquiring a nonnull imaginary part, which implies a dynamical transition from nonoscillatory to oscillatory linear response to perturbations. Lastly, we ascertain the discontinuous nature of this transition.

cond-mat.dis-nn↗