arXiv subjects

Fernando L. Metz

Publications and source records attributed to Fernando L. Metz.

At least 19 recordsLinked to original sources

arXiv2026

Dynamical cavity method for continuous-time complex systems on sparse random graphs

Dynamical mean-field theory (DMFT) reduces dense high-dimensional disordered dynamics to a self-consistent effective stochastic process. For sparse and heterogeneous networks, however, local fields contain finitely many strong inputs, so the Gaussian closure mechanisms of dense DMFT need not apply. We develop a continuous-time cavity derivation of sparse-network DMFT at the level of path measures for stochastic dynamics with general pairwise interactions on sparse random graphs. The cavity equations are exact on trees and yield the finite-time thermodynamic description on locally tree-like graphs. They show explicitly how reciprocity changes dynamical closure: fully directed graphs recover the sparse directed path-probability equation, whereas reciprocal or bidirected edges require conditional path kernels driven by the imposed history of the receiving node. Ensemble averaging gives laws over path-probability messages, with barycenters and higher-message moments closing by multilinearity and independence of incoming branches. A causal discrete-time derivation yields the corresponding population-dynamics representation, distinguishing trajectory populations for directed graphs from conditional branch-law or finite-depth tree populations for reciprocal graphs. We also formulate finite-memory numerical closures and test them in an additive-input recurrent neural network specialization. Finally, high-connectivity limits are obtained as projections of the sparse path-measure theory, clarifying when dense drift, noise, and response channels reduce to standard low-dimensional DMFT and when path-level descriptions remain essential.

cond-mat.dis-nn↗

arXiv2026

BBP transition and the leading eigenvector of the spiked Wigner model with inhomogeneous noise

The spiked Wigner ensemble is a prototypical model for high-dimensional inference. We study the spectral properties of an inhomogeneous rank-one spiked Wigner model in which the variance of each entry of the noise matrix is itself a random variable. In the high-dimensional limit, we derive exact equations for the spectral edges, the outlier eigenvalue, and the distribution of the components of the outlier eigenvector. These equations determine the BBP transition line that separates the gapped phase, where the signal is detectable, from the gapless phase. In the gapped regime, the distribution of the outlier eigenvector provides a natural estimator of the spike. We solve the equations for a noise matrix whose variances are generated from a truncated power-law distribution. In this case, the BBP transition line is non-monotonic, showing that an inhomogeneous noise can enhance signal detectability.

cond-mat.dis-nn↗

arXiv2025

Zero-temperature dynamics of the spherical model with non-reciprocal interactions

We analytically solve the zero-temperature dynamics of the spherical model with non-reciprocal random interactions drawn from the real elliptic ensemble of random matrices, where a single parameter $\eta$ continuously interpolates between purely symmetric ($\eta=1$) and purely antisymmetric ($\eta=-1$) couplings. We show that the two-time correlation and response functions depend on both times in the presence of non-reciprocal interactions, reflecting the breakdown of time-translation invariance and the absence of equilibrium at long times. Nevertheless, the long-time relaxation of the two-time observables is governed by exponential decays, in contrast to the slow, power-law relaxation characteristic of the model with purely symmetric interactions. We further show that, when the interactions present antisymmetric correlations of strength $\eta <0$, there is a time scale $\tau(\eta)$ above which the dynamics undergoes a transition to an oscillatory regime where the two-time observables display periodic oscillations with an exponentially decaying amplitude. Overall, our results give a detailed account of the dynamics of the spherical model with non-reciprocal interactions at zero temperature, providing a benchmark for the study of complex systems with nonlinear and asymmetric interactions.

cond-mat.stat-mech↗

arXiv2025

Epidemic threshold and localization of the SIS model on directed complex networks

We study the susceptible-infected-susceptible (SIS) model on directed complex networks within the quenched mean-field approximation. Combining results from random matrix theory with an analytic approach to the distribution of fixed-point infection probabilities, we derive the phase diagram and show that the model exhibits a nonequilibrium phase transition between the absorbing and endemic phases for $c \geq \lambda^{-1}$, where $c$ is the mean degree and $\lambda$ the average infection rate. Interestingly, the critical line is independent of the degree distribution but is highly sensitive to the form of the infection-rate distribution. We further show that the inverse participation ratio of infection probabilities diverges near the epidemic threshold, indicating that the disease may become localized on a small fraction of nodes. These results provide a systematic characterization of how network heterogeneities shape epidemic spreading on directed contact networks within the quenched mean-field approximation.

physics.soc-ph↗

arXiv2025

Effects of clustering heterogeneity on the spectral density of sparse networks

We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effect of clustering on the spectral properties of the network adjacency matrix. In the case of heterogeneous networks, we demonstrate that the spectral density becomes more symmetric as the fluctuations in the triangle-degree sequence increase. This phenomenon is explained by the small clustering coefficient of networks with a large variance of the triangle-degree distribution. In the homogeneous case of regular clustered networks, we find that both perturbative and non-perturbative approximations fail to predict the spectral density in the high-connectivity limit. This suggests that traditional large-degree approximations may be ineffective in studying the spectral properties of networks with more complex motifs. Our theoretical results are fully confirmed by numerical diagonalizations of finite adjacency matrices.

cond-mat.dis-nn↗

arXiv2025

Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks

Although real-world complex systems typically interact through sparse and heterogeneous networks, analytic solutions of their dynamics are limited to models with all-to-all interactions. Here, we solve the dynamics of a broad range of nonlinear models of complex systems on sparse directed networks with a random structure. By generalizing dynamical mean-field theory to sparse systems, we derive an exact equation for the path-probability describing the effective dynamics of a single degree of freedom. Our general solution applies to key models in the study of neural networks, ecosystems, epidemic spreading, and synchronization. Using the population dynamics algorithm, we solve the path-probability equation to determine the phase diagram of a seminal neural network model in the sparse regime, showing that this model undergoes a transition from a fixed-point phase to chaos as a function of the network topology.

cond-mat.dis-nn↗

arXiv2024

Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks

We study the spectral properties and eigenvector statistics of the Laplacian on highly-connected networks with random coupling strengths and a gamma distribution of rescaled degrees. The spectral density, the distribution of the local density of states, the singularity spectrum and the multifractal exponents of this model exhibit a rich behaviour as a function of the first two moments of the coupling strengths and the variance of the rescaled degrees. In the case of random coupling strengths, the spectral density diverges within the bulk of the spectrum when degree fluctuations are strong enough. The emergence of this singular behaviour marks a transition from non-ergodic delocalized states to localized eigenvectors that exhibit pronounced multifractal scaling. For constant coupling strengths, the bulk of the spectrum is characterized by a regular spectral density. In this case, the corresponding eigenvectors display localization properties reminiscent of the critical point of the Anderson localization transition on random graphs.

cond-mat.dis-nn↗

arXiv2024

Statistical properties of speckle patterns for a random number of scatterers and nonuniform phase distributions

The statistical properties of speckle patterns have important applications in optics, oceanography, and transport phenomena in disordered systems. Here we obtain closed-form analytic results for the amplitude distribution of speckle patterns formed by a random number of partial waves characterized by an arbitrary phase distribution, generalizing classical results of the random walk theory of speckle patterns. We show that the functional form of the amplitude distribution is solely determined by the distribution of the number of scatterers, while the phase distribution only influences the scale parameters. In the case of a non-random number of scatterers, we find an analytic expression for the amplitude distribution that extends the Rayleigh law to non-uniform random phases. For a negative binomial distribution of the number of scatterers, our results reveal that large fluctuations of the wave amplitudes become more pronounced in the case of biased random phases. We present numerical results that fully support our analytic findings.

physics.optics↗

arXiv2023

Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems.

cond-mat.stat-mech↗

arXiv2022

Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties

The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random graphs with an arbitrary degree distribution in the high-connectivity limit, from which we perform a thorough analysis of the impact of degree fluctuations on the spectral density, the inverse participation ratio, and the distribution of the local density of states. We show that all eigenvectors are extended and that the spectral density exhibits a logarithmic or a power-law divergence when the variance of the degree distribution is large enough. We elucidate this singular behaviour by showing that the distribution of the local density of states at the center of the spectrum displays a power-law tail determined by the variance of the degree distribution. In the regime of weak degree fluctuations the spectral density has a finite support, which promotes the stability of large complex systems on random graphs.

cond-mat.dis-nn↗

arXiv2022

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington-Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree $c$ scales as a power $c \propto N^{b}$ ($b < 1$) of the total number $N \gg 1$ of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

cond-mat.dis-nn↗

arXiv2021

Analytic solution of the two-star model with correlated degrees

Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime. The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors). We compute exactly the network free energy and show that this model undergoes a first-order transition to a condensed phase. For non-negative degree correlations between next nearest neighbors, the degree distribution inside the condensed phase has a single peak at the largest degree, while for negative degree correlations between next nearest neighbors the condensed phase is characterized by a bimodal degree distribution. We calculate the degree assortativities and show they are non-monotonic functions of the model parameters, with a discontinuous behavior at the first-order transition. The first-order critical line terminates at a second-order critical point, whose location in the phase diagram can be accurately determined. Our results can help to develop more detailed models of complex networks with correlated degrees.

cond-mat.stat-mech↗

arXiv2021

Localization and universality of eigenvectors in directed random graphs

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree fluctuations. For delocalized eigenvectors, we recover the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.

cond-mat.dis-nn↗

arXiv2020

The spectral density of dense random networks and the breakdown of the Wigner semicircle law

Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the configuration model of networks with four distinct degree distributions, we show that the spectral density of the adjacency matrices of dense random networks is determined by the strength of the degree fluctuations. In particular, the eigenvalue distribution of dense networks with an exponential degree distribution is governed by a simple equation, from which we uncover a logarithmic singularity in the spectral density. We also derive a relation between the fourth moment of the eigenvalue distribution and the variance of the degree distribution, which leads to a sufficient condition for the breakdown of the Wigner semicircle law for dense random networks. Based on the same relation, we propose a classification scheme of the distinct universal behaviours of the spectral density in the dense limit. Our theoretical findings should lead to important insights on the mean-field behaviour of models defined on graphs.

cond-mat.dis-nn↗

arXiv2020

Analytic approach for the number statistics of non-Hermitian random matrices

We introduce a powerful analytic method to study the statistics of the number $\mathcal{N}_{\textbf{A}}(γ)$ of eigenvalues inside any contour $γ\in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf A}$. Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable. The main outcome is an effective theory that determines the cumulant generating function of $\mathcal{N}_{\textbf{A}}$ via a path integral along $γ$, with the path probability distribution following from the solution of a self-consistent equation. We derive the expressions for the mean and the variance of $\mathcal{N}_{\textbf{A}}$ as well as for the rate function governing rare fluctuations of ${\mathcal{N}}_{\textbf{A}}{(γ)}$. All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.

cond-mat.dis-nn↗

arXiv2019

Superextreme Waves Generation in the Linear Regime

Extreme or rogue waves are large and unexpected waves appearing with higher probability than predicted by Gaussian statistics. Although their formation is explained by both linear and nonlinear wave propagation, nonlinearity has been considered a necessary ingredient to generate superextreme waves, i.e., an enhanced wave amplification, where the wave amplitudes exceed by far those of standard rogue waves. Here we show, experimentally and theoretically, that superextreme optical waves emerge in the simple case of linear one-dimensional light diffraction. The underlying physics is a long-range correlation on the random initial phases of the light waves. When subgroups of random phases appear recurrently along the spatial phase distribution, a more ordered phase structure greatly increases the probability of constructive interference to generate superextreme events, i.e., non-Gaussian statistics with super-long tails. Our results consist of a significant advance in the understanding of extreme waves formation by linear superposition of random waves, with applications in a large variety of wave systems.

physics.optics↗

arXiv2019

Phase transitions in atypical systems induced by a condensation transition on graphs

Random graphs undergo structural phase transitions that are crucial for dynamical processes and cooperative behavior of models defined on graphs. In this work we investigate the impact of a first-order structural transition on the thermodynamics of the Ising model defined on Erdös-Rényi random graphs, as well as on the eigenvalue distribution of the adjacency matrix of the same graphical model. The structural transition in question yields graph samples exhibiting condensation, characterized by a large number of nodes having degrees in a narrow interval. We show that this condensation transition induces distinct thermodynamic first-order transitions between the paramagnetic and the ferromagnetic phases of the Ising model. The condensation transition also leads to an abrupt change in the global eigenvalue statistics of the adjacency matrix, which renders the second moment of the eigenvalue distribution discontinuous. As a side result, we derive the critical line determining the percolation transition in Erdös-Rényi graph samples that feature condensation of degrees.

cond-mat.dis-nn↗

arXiv2019

Condensation of degrees emerging through a first-order phase transition in classical random graphs

Due to their conceptual and mathematical simplicity, Erdös-Rényi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a first-order phase transition between a Poisson-like distribution and a condensed phase, the latter characterized by a large fraction of nodes having degrees in a limited sector of their configuration space. The mechanism underlying the first-order transition is discussed in light of standard concepts in statistical physics. We uncover the phase diagram characterizing the ensemble space of the model and we evaluate the rate function governing the probability to observe a condensed state, which shows that condensation of degrees is a rare statistical event akin to similar condensation phenomena recently observed in several other systems. Monte Carlo simulations confirm the exactness of our theoretical results.

cond-mat.dis-nn↗