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Dynamical cavity method for continuous-time complex systems on sparse random graphs
Authors:
Fernando L. Metz,
Isaac Pérez Castillo
Abstract:
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…
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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.
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Submitted 7 June, 2026;
originally announced June 2026.
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BBP transition and the leading eigenvector of the spiked Wigner model with inhomogeneous noise
Authors:
Leonardo S. Ferreira,
Fernando L. Metz
Abstract:
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…
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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.
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Submitted 21 July, 2026; v1 submitted 20 April, 2026;
originally announced April 2026.
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Zero-temperature dynamics of the spherical model with non-reciprocal interactions
Authors:
Daniel A. Stariolo,
Fernando L. Metz
Abstract:
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 $η$ continuously interpolates between purely symmetric ($η=1$) and purely antisymmetric ($η=-1$) couplings. We show that the two-time correlation and response functions depend on both times in the prese…
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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 $η$ continuously interpolates between purely symmetric ($η=1$) and purely antisymmetric ($η=-1$) 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 $η<0$, there is a time scale $τ(η)$ 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.
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Submitted 31 March, 2026; v1 submitted 20 November, 2025;
originally announced November 2025.
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Epidemic threshold and localization of the SIS model on directed complex networks
Authors:
Vinícius B. Müller,
Fernando L. Metz
Abstract:
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 f…
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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 $c \geq λ^{-1}$, where $c$ is the mean degree and $λ$ 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.
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Submitted 11 December, 2025; v1 submitted 8 August, 2025;
originally announced August 2025.
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Simulating continuous-space systems with quantum-classical wave functions
Authors:
Friederike Metz,
Gabriel Pescia,
Giuseppe Carleo
Abstract:
Most non-relativistic interacting quantum many-body systems, such as atomic and molecular ensembles or materials, are naturally described in terms of continuous-space Hamiltonians. The simulation of their ground-state properties on digital quantum computers is challenging because current algorithms require discretization, which usually amounts to choosing a finite basis set, inevitably introducing…
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Most non-relativistic interacting quantum many-body systems, such as atomic and molecular ensembles or materials, are naturally described in terms of continuous-space Hamiltonians. The simulation of their ground-state properties on digital quantum computers is challenging because current algorithms require discretization, which usually amounts to choosing a finite basis set, inevitably introducing errors. In this work, we propose an alternative, discretization-free approach that combines classical and quantum resources in a global variational ansatz, optimized using the framework of variational Monte Carlo. We introduce both purely quantum as well as hybrid quantum-classical ansatze and benchmark them on three paradigmatic continuous-space systems that are either very challenging or beyond the reach of current quantum approaches: the one-dimensional quantum rotor model, a system of Helium-3 particles in one and two dimensions, and the two-dimensional homogeneous electron gas. We embed relevant constraints such as the antisymmetry of fermionic wave functions directly into the ansatz. Many-body correlations are introduced via backflow transformations represented by parameterized quantum circuits. We demonstrate that the accuracy of the simulation can be systematically improved by increasing the number of circuit parameters and study the effects of shot noise. Furthermore, we show that the hybrid ansatz improves the ground-state energies obtained using the purely classical wave function.
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Submitted 10 September, 2024;
originally announced September 2024.
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Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures
Authors:
Leonardo Ferreira,
Fernando Metz,
Paolo Barucca
Abstract:
We introduce a random matrix model for the stationary covariance of multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, where the covariance is constrained by the Sylvester-Lyapunov equation. Using the replica method, we compute the spectral density of the equal-time covariance matrix characterizing the stationary states, demonstrating that this model undergoes a transition…
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We introduce a random matrix model for the stationary covariance of multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, where the covariance is constrained by the Sylvester-Lyapunov equation. Using the replica method, we compute the spectral density of the equal-time covariance matrix characterizing the stationary states, demonstrating that this model undergoes a transition between stable and unstable states. In the stable regime, the spectral density has a finite and positive support, whereas negative eigenvalues emerge in the unstable regime. We determine the critical line separating these regimes and show that the spectral density exhibits a power-law tail at marginal stability, with an exponent independent of the temperature distribution. Additionally, we compute the spectral density of the lagged covariance matrix characterizing the stationary states of linear transformations of the original dynamical variables. Our random-matrix model is potentially interesting to understand the spectral properties of empirical correlation matrices appearing in the study of complex systems.
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Submitted 2 September, 2024;
originally announced September 2024.
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Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks
Authors:
Jeferson D. da Silva,
Diego Tapias,
Peter Sollich,
Fernando L. Metz
Abstract:
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 coupl…
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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.
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Submitted 11 December, 2024; v1 submitted 23 August, 2024;
originally announced August 2024.
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Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks
Authors:
Fernando L. Metz
Abstract:
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…
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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.
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Submitted 21 January, 2025; v1 submitted 10 June, 2024;
originally announced June 2024.
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Effects of clustering heterogeneity on the spectral density of sparse networks
Authors:
Tuan Minh Pham,
Thomas Peron,
Fernando L. Metz
Abstract:
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 t…
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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.
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Submitted 28 January, 2025; v1 submitted 11 April, 2024;
originally announced April 2024.
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Statistical properties of speckle patterns for a random number of scatterers and nonuniform phase distributions
Authors:
Fernando L. Metz,
Cristian Bonatto,
Sandra D. Prado
Abstract:
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 speck…
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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.
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Submitted 10 January, 2024; v1 submitted 11 October, 2023;
originally announced October 2023.
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Overhead-constrained circuit knitting for variational quantum dynamics
Authors:
Gian Gentinetta,
Friederike Metz,
Giuseppe Carleo
Abstract:
Simulating the dynamics of large quantum systems is a formidable yet vital pursuit for obtaining a deeper understanding of quantum mechanical phenomena. While quantum computers hold great promise for speeding up such simulations, their practical application remains hindered by limited scale and pervasive noise. In this work, we propose an approach that addresses these challenges by employing circu…
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Simulating the dynamics of large quantum systems is a formidable yet vital pursuit for obtaining a deeper understanding of quantum mechanical phenomena. While quantum computers hold great promise for speeding up such simulations, their practical application remains hindered by limited scale and pervasive noise. In this work, we propose an approach that addresses these challenges by employing circuit knitting to partition a large quantum system into smaller subsystems that can each be simulated on a separate device. The evolution of the system is governed by the projected variational quantum dynamics (PVQD) algorithm, supplemented with constraints on the parameters of the variational quantum circuit, ensuring that the sampling overhead imposed by the circuit knitting scheme remains controllable. We test our method on quantum spin systems with multiple weakly entangled blocks each consisting of strongly correlated spins, where we are able to accurately simulate the dynamics while keeping the sampling overhead manageable. Further, we show that the same method can be used to reduce the circuit depth by cutting long-ranged gates.
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Submitted 20 March, 2024; v1 submitted 14 September, 2023;
originally announced September 2023.
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Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise
Authors:
Leonardo S. Ferreira,
Fernando L. Metz
Abstract:
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 nonequili…
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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.
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Submitted 20 March, 2023; v1 submitted 19 December, 2022;
originally announced December 2022.
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Analytic solution of the resolvent equations for heterogeneous random graphs: spectral and localization properties
Authors:
Jeferson D. Silva,
Fernando L. Metz
Abstract:
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 s…
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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.
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Submitted 14 September, 2022;
originally announced September 2022.
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Self-Correcting Quantum Many-Body Control using Reinforcement Learning with Tensor Networks
Authors:
Friederike Metz,
Marin Bukov
Abstract:
Quantum many-body control is a central milestone en route to harnessing quantum technologies. However, the exponential growth of the Hilbert space dimension with the number of qubits makes it challenging to classically simulate quantum many-body systems and consequently, to devise reliable and robust optimal control protocols. Here, we present a novel framework for efficiently controlling quantum…
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Quantum many-body control is a central milestone en route to harnessing quantum technologies. However, the exponential growth of the Hilbert space dimension with the number of qubits makes it challenging to classically simulate quantum many-body systems and consequently, to devise reliable and robust optimal control protocols. Here, we present a novel framework for efficiently controlling quantum many-body systems based on reinforcement learning (RL). We tackle the quantum control problem by leveraging matrix product states (i) for representing the many-body state and, (ii) as part of the trainable machine learning architecture for our RL agent. The framework is applied to prepare ground states of the quantum Ising chain, including states in the critical region. It allows us to control systems far larger than neural-network-only architectures permit, while retaining the advantages of deep learning algorithms, such as generalizability and trainable robustness to noise. In particular, we demonstrate that RL agents are capable of finding universal controls, of learning how to optimally steer previously unseen many-body states, and of adapting control protocols on-the-fly when the quantum dynamics is subject to stochastic perturbations. Furthermore, we map the QMPS framework to a hybrid quantum-classical algorithm that can be performed on noisy intermediate-scale quantum devices and test it under the presence of experimentally relevant sources of noise.
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Submitted 11 May, 2023; v1 submitted 27 January, 2022;
originally announced January 2022.
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Mean-field theory of vector spin models on networks with arbitrary degree distributions
Authors:
Fernando L. Metz,
Thomas Peron
Abstract:
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…
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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.
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Submitted 9 February, 2022; v1 submitted 21 October, 2021;
originally announced October 2021.
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Analytic solution of the two-star model with correlated degrees
Authors:
Maíra Bolfe,
Fernando L. Metz,
Edgar Guzmán-González,
Isaac Pérez Castillo
Abstract:
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 fr…
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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.
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Submitted 3 August, 2021; v1 submitted 18 February, 2021;
originally announced February 2021.
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Deep learning based quantum vortex detection in atomic Bose-Einstein condensates
Authors:
Friederike Metz,
Juan Polo,
Natalya Weber,
Thomas Busch
Abstract:
Quantum vortices naturally emerge in rotating Bose-Einstein condensates (BECs) and, similarly to their classical counterparts, allow the study of a range of interesting out-of-equilibrium phenomena like turbulence and chaos. However, the study of such phenomena requires to determine the precise location of each vortex within a BEC, which becomes challenging when either only the condensate density…
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Quantum vortices naturally emerge in rotating Bose-Einstein condensates (BECs) and, similarly to their classical counterparts, allow the study of a range of interesting out-of-equilibrium phenomena like turbulence and chaos. However, the study of such phenomena requires to determine the precise location of each vortex within a BEC, which becomes challenging when either only the condensate density is available or sources of noise are present, as is typically the case in experimental settings. Here, we introduce a machine learning based vortex detector motivated by state-of-the-art object detection methods that can accurately locate vortices in simulated BEC density images. Our model allows for robust and real-time detection in noisy and non-equilibrium configurations. Furthermore, the network can distinguish between vortices and anti-vortices if the condensate phase profile is also available. We anticipate that our vortex detector will be advantageous both for experimental and theoretical studies of the static and dynamical properties of vortex configurations in BECs.
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Submitted 13 April, 2021; v1 submitted 23 December, 2020;
originally announced December 2020.
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The spectral density of dense random networks and the breakdown of the Wigner semicircle law
Authors:
Fernando L. Metz,
Jeferson D. Silva
Abstract:
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 fluct…
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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.
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Submitted 22 October, 2020; v1 submitted 29 July, 2020;
originally announced July 2020.
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Localization and universality of eigenvectors in directed random graphs
Authors:
Fernando L. Metz,
Izaak Neri
Abstract:
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 exac…
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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.
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Submitted 10 March, 2021; v1 submitted 27 July, 2020;
originally announced July 2020.
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Analytic approach for the number statistics of non-Hermitian random matrices
Authors:
Antonio Tonatiúh Ramos Sánchez,
Edgar Guzmán-González,
Isaac Pérez Castillo,
Fernando L. Metz
Abstract:
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 illustr…
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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.
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Submitted 20 July, 2020;
originally announced July 2020.
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Phase transitions in atypical systems induced by a condensation transition on graphs
Authors:
Edgar Guzmán-González,
Isaac Pérez Castillo,
Fernando L. Metz
Abstract:
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. Th…
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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.
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Submitted 23 September, 2019;
originally announced September 2019.
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Linear stability analysis for large dynamical systems on directed random graphs
Authors:
Izaak Neri,
Fernando Lucas Metz
Abstract:
We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is sur…
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We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.
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Submitted 3 July, 2026; v1 submitted 19 August, 2019;
originally announced August 2019.
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Condensation of degrees emerging through a first-order phase transition in classical random graphs
Authors:
Fernando L. Metz,
Isaac Pérez Castillo
Abstract:
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 grap…
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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.
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Submitted 20 April, 2019;
originally announced April 2019.
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Spectral Theory of Sparse Non-Hermitian Random Matrices
Authors:
Fernando Lucas Metz,
Izaak Neri,
Tim Rogers
Abstract:
Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry…
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Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples -- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erdős-Rényi graphs, and adjacency matrices of weighted oriented Erdős-Rényi graphs -- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.
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Submitted 20 February, 2024; v1 submitted 26 November, 2018;
originally announced November 2018.
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Two-leg ladder Bose Hubbard models with staggered fluxes
Authors:
Rashi Sachdeva,
Friederike Metz,
Manpreet Singh,
Tapan Mishra,
Thomas Busch
Abstract:
We investigate the ground state properties of ultracold atoms trapped in a two-leg ladder potential in the presence of an artificial magnetic field in a staggered configuration. We focus on the strongly interacting regime and use the Landau theory of phase transitions and a mean field Gutzwiller variational method to identify the stable superfluid phases and their boundaries with the Mott-insulato…
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We investigate the ground state properties of ultracold atoms trapped in a two-leg ladder potential in the presence of an artificial magnetic field in a staggered configuration. We focus on the strongly interacting regime and use the Landau theory of phase transitions and a mean field Gutzwiller variational method to identify the stable superfluid phases and their boundaries with the Mott-insulator regime as a function of magnetic flux. In addition, we calculate the local and chiral currents of these superfluid phases, which show a staggered vortex anti-vortex configuration. The analytical results are confirmed by numerical simulations using a cluster mean-field theory approach.
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Submitted 16 August, 2018;
originally announced August 2018.
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Phase diagram and metastability of the Ising model on two coupled networks
Authors:
Maíra Bolfe,
Lucas Nicolao,
Fernando L. Metz
Abstract:
We explore the cooperative behaviour and phase transitions of interacting networks by studying a simplified model consisting of Ising spins placed on the nodes of two coupled Erdös-Rényi random graphs. We derive analytical expressions for the free-energy of the system and the magnetization of each graph, from which the phase diagrams, the stability of the different states, and the nature of the tr…
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We explore the cooperative behaviour and phase transitions of interacting networks by studying a simplified model consisting of Ising spins placed on the nodes of two coupled Erdös-Rényi random graphs. We derive analytical expressions for the free-energy of the system and the magnetization of each graph, from which the phase diagrams, the stability of the different states, and the nature of the transitions among them, are clearly characterized. We show that a metastable state appears discontinuously by varying the model parameters, yielding a region in the phase diagram where two solutions coexist. By performing Monte-Carlo simulations, we confirm the exactness of our main theoretical results and show that the typical time the system needs to escape from a metastable state grows exponentially fast as a function of the temperature, characterizing ergodicity breaking in the thermodynamic limit.
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Submitted 23 August, 2018; v1 submitted 4 April, 2018;
originally announced April 2018.
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Theory for the conditioned spectral density of non-invariant random matrices
Authors:
Isaac Pérez Castillo,
Fernando L. Metz
Abstract:
We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution conditioned to have a fixed fraction $k$ of eigenvalues smaller than $x \in \mathbb{R}$, provides the spectrum of random matrix samples that deviate atypically from th…
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We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution conditioned to have a fixed fraction $k$ of eigenvalues smaller than $x \in \mathbb{R}$, provides the spectrum of random matrix samples that deviate atypically from the average behavior. We apply our theory to sparse random matrices and unveil strikingly new and generic properties, namely: (i) their conditioned spectral density has compact support; (ii) it does not experience any abrupt transition for $k$ around its typical value; (iii) its eigenvalues do not accumulate at $x$. Moreover, our work points towards other types of transitions in the conditioned spectral density for values of $k$ away from its typical value. These properties follow from the weak or absent eigenvalue repulsion in sparse ensembles and they are in sharp contrast to those displayed by classic or rotationally invariant random matrices. The exactness of our theoretical findings are confirmed through numerical diagonalization of finite random matrices.
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Submitted 13 August, 2018; v1 submitted 8 March, 2018;
originally announced March 2018.
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Large deviation theory for diluted Wishart random matrices
Authors:
Isaac Pérez Castillo,
Fernando L. Metz
Abstract:
Wishart random matrices with a sparse or diluted structure are ubiquitous in the processing of large datasets, with applications in physics, biology and economy. In this work we develop a theory for the eigenvalue fluctuations of diluted Wishart random matrices, based on the replica approach of disordered systems. We derive an analytical expression for the cumulant generating function of the numbe…
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Wishart random matrices with a sparse or diluted structure are ubiquitous in the processing of large datasets, with applications in physics, biology and economy. In this work we develop a theory for the eigenvalue fluctuations of diluted Wishart random matrices, based on the replica approach of disordered systems. We derive an analytical expression for the cumulant generating function of the number of eigenvalues $\mathcal{I}_N(x)$ smaller than $x\in\mathbb{R}^{+}$, from which all cumulants of $\mathcal{I}_N(x)$ and the rate function $Ψ_{x}(k)$ controlling its large deviation probability $\text{Prob}[\mathcal{I}_N(x)=kN] \asymp e^{-NΨ_{x}(k)}$ follow. Explicit results for the mean value and the variance of $\mathcal{I}_N(x)$, its rate function, and its third cumulant are discussed and thoroughly compared to numerical diagonalization, showing a very good agreement. The present work establishes the theoretical framework put forward in a recent letter [Phys. Rev. Lett. 117, 104101] as an exact and compelling approach to deal with eigenvalue fluctuations of sparse random matrices.
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Submitted 19 March, 2018; v1 submitted 11 January, 2018;
originally announced January 2018.
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Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
Authors:
Fernando L. Metz,
Isaac Pérez Castillo
Abstract:
We calculate the level compressibility $χ(W,L)$ of the energy levels inside $[-L/2,L/2]$ for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in $[-W/2,W/2]$. We show that $χ(W,L)$ approaches the limit $\lim_{L \rightarrow 0^+} χ(W,L) = 0$ for a broad interval of the disorder strength $W$ within the extended phase, including the region of…
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We calculate the level compressibility $χ(W,L)$ of the energy levels inside $[-L/2,L/2]$ for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in $[-W/2,W/2]$. We show that $χ(W,L)$ approaches the limit $\lim_{L \rightarrow 0^+} χ(W,L) = 0$ for a broad interval of the disorder strength $W$ within the extended phase, including the region of $W$ close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erdös-Rényi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.
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Submitted 10 August, 2017; v1 submitted 30 March, 2017;
originally announced March 2017.
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Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
Authors:
Izaak Neri,
Fernando Lucas Metz
Abstract:
Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical processes. We present a general and exact theory for the eigenvalue outliers of random matrices with a local tree structure. For adjacency and Laplacian matrice…
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Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical processes. We present a general and exact theory for the eigenvalue outliers of random matrices with a local tree structure. For adjacency and Laplacian matrices of oriented random graphs, we derive analytical expressions for the eigenvalue outliers, the first moments of the distribution of eigenvector elements associated with an outlier, the support of the spectral density, and the spectral gap. We show that these spectral observables obey universal expressions, which hold for a broad class of oriented random matrices.
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Submitted 23 November, 2016;
originally announced November 2016.
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Large deviation function for the number of eigenvalues of sparse random graphs inside an interval
Authors:
Fernando L. Metz,
Isaac Pérez Castillo
Abstract:
We present a general method to obtain the exact rate function $Ψ_{[a,b]}(k)$ controlling the large deviation probability $\text{Prob}[\mathcal{I}_N[a,b]=kN] \asymp e^{-NΨ_{[a,b]}(k)}$ that a $N \times N$ sparse random matrix has $\mathcal{I}_N[a,b]=kN$ eigenvalues inside the interval $[a,b]$. The method is applied to study the eigenvalue statistics in two distinct examples: (i) the shifted index n…
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We present a general method to obtain the exact rate function $Ψ_{[a,b]}(k)$ controlling the large deviation probability $\text{Prob}[\mathcal{I}_N[a,b]=kN] \asymp e^{-NΨ_{[a,b]}(k)}$ that a $N \times N$ sparse random matrix has $\mathcal{I}_N[a,b]=kN$ eigenvalues inside the interval $[a,b]$. The method is applied to study the eigenvalue statistics in two distinct examples: (i) the shifted index number of eigenvalues for an ensemble of Erdös-Rényi graphs and (ii) the number of eigenvalues within a bounded region of the spectrum for the Anderson model on regular random graphs. A salient feature of the rate function in both cases is that, unlike rotationally invariant random matrices, it is asymmetric with respect to its minimum. The asymmetric character depends on the disorder in a way that is compatible with the distinct eigenvalue statistics corresponding to localized and delocalized eigenstates. The results also show that the number variance $σ_{N}^{2}$ for the Anderson model on a regular graph scales as $σ_{N}^{2} \propto N$ ($N \gg 1$) for any nonzero disorder, which is consistent with the absence of level-repulsion in the extended phase. Our theoretical findings are thoroughly compared to numerical diagonalization in both cases, showing a reasonable good agreement.
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Submitted 2 September, 2016; v1 submitted 18 March, 2016;
originally announced March 2016.
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Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices
Authors:
Fernando L. Metz
Abstract:
We discuss an approach to compute the first and second moments of the number of eigenvalues $I_N$ that lie in an arbitrary interval of the real line for $N \times N$ Gaussian random matrices. The method combines the standard replica-symmetric theory with a perturbative expansion of the saddle-point action up to $O(1/N)$ ($N \gg 1$), leading to the correct logarithmic scaling of the variance…
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We discuss an approach to compute the first and second moments of the number of eigenvalues $I_N$ that lie in an arbitrary interval of the real line for $N \times N$ Gaussian random matrices. The method combines the standard replica-symmetric theory with a perturbative expansion of the saddle-point action up to $O(1/N)$ ($N \gg 1$), leading to the correct logarithmic scaling of the variance $\langle I_{N}^{2} \rangle - \langle I_N \rangle^2 = O(\ln N)$ as well as to an analytical expression for the $O(1/N)$ correction to the average $\langle I_N \rangle/N$. Standard results for the number variance at the local scaling regime are recovered in the limit of a vanishing interval. The limitations of the replica-symmetric method are unveiled by comparing our results with those derived through exact methods. The present work represents an important step to study the fluctuations of $I_N$ in non-invariant random matrix ensembles, where the joint distribution of eigenvalues is not known.
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Submitted 20 November, 2017; v1 submitted 22 October, 2015;
originally announced October 2015.
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Index statistical properties of sparse random graphs
Authors:
Fernando L. Metz,
Daniel A. Stariolo
Abstract:
Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $\mathcal{P}_N(K,λ)$ that a large $N \times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $λ$. The method allows to determine, in principle, all moments of $\mathcal{P}_N(K,λ)$, from which the typical sample to sample fluctuations can be fully c…
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Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $\mathcal{P}_N(K,λ)$ that a large $N \times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $λ$. The method allows to determine, in principle, all moments of $\mathcal{P}_N(K,λ)$, from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with $N \gg 1$ for $|λ| > 0$, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of $λ$. These results contrast with rotationally invariant random matrices, where the index variance scales only as $\ln N$, with an universal prefactor that is independent of $λ$. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.
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Submitted 4 September, 2015;
originally announced September 2015.
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Statistical mechanics of the spherical hierarchical model with random fields
Authors:
Fernando L. Metz,
Jacopo Rocchi,
Pierfrancesco Urbani
Abstract:
We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding $D$-dimensional short-range model, leading to co…
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We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding $D$-dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and $D$-dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.
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Submitted 5 June, 2014;
originally announced June 2014.
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Finite size correction to the spectrum of regular random graphs: an analytical solution
Authors:
Fernando L. Metz,
Giorgio Parisi,
Luca Leuzzi
Abstract:
We develop a thorough analytical study of the $O(1/N)$ correction to the spectrum of regular random graphs with $N \rightarrow \infty$ nodes. The finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the $O(1/N)$ correcti…
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We develop a thorough analytical study of the $O(1/N)$ correction to the spectrum of regular random graphs with $N \rightarrow \infty$ nodes. The finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the $O(1/N)$ correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.
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Submitted 17 November, 2014; v1 submitted 11 March, 2014;
originally announced March 2014.
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The renormalization flow of the hierarchical Anderson model at weak disorder
Authors:
F. L. Metz,
L. Leuzzi,
G. Parisi
Abstract:
We study the flow of the renormalized model parameters obtained from a sequence of simple transformations of the 1D Anderson model with long-range hierarchical hopping. Combining numerical results with a perturbative approach for the flow equations, we identify three qualitatively different regimes at weak disorder. For a sufficiently fast decay of the hopping energy, the Cauchy distribution is th…
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We study the flow of the renormalized model parameters obtained from a sequence of simple transformations of the 1D Anderson model with long-range hierarchical hopping. Combining numerical results with a perturbative approach for the flow equations, we identify three qualitatively different regimes at weak disorder. For a sufficiently fast decay of the hopping energy, the Cauchy distribution is the only stable fixed-point of the flow equations, whereas for sufficiently slowly decaying hopping energy the renormalized parameters flow to a delta peak fixed-point distribution. In an intermediate range of the hopping decay, both fixed-point distributions are stable and the stationary solution is determined by the initial configuration of the random parameters. We present results for the critical decay of the hopping energy separating the different regimes.
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Submitted 4 November, 2013;
originally announced November 2013.
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Transition between localized and extended states in the hierarchical Anderson model
Authors:
F. L. Metz,
L. Leuzzi,
G. Parisi,
V. Sacksteder IV
Abstract:
We present strong numerical evidence for the existence of a localization-delocalization transition in the eigenstates of the 1-D Anderson model with long-range hierarchical hopping. Hierarchical models are important because of the well-known mapping between their phases and those of models with short range hopping in higher dimensions, and also because the renormalization group can be applied exac…
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We present strong numerical evidence for the existence of a localization-delocalization transition in the eigenstates of the 1-D Anderson model with long-range hierarchical hopping. Hierarchical models are important because of the well-known mapping between their phases and those of models with short range hopping in higher dimensions, and also because the renormalization group can be applied exactly without the approximations that generally are required in other models. In the hierarchical Anderson model we find a finite critical disorder strength Wc where the average inverse participation ratio goes to zero; at small disorder W < Wc the model lies in a delocalized phase. This result is based on numerical calculation of the inverse participation ratio in the infinite volume limit using an exact renormalization group approach facilitated by the model's hierarchical structure. Our results are consistent with the presence of an Anderson transition in short-range models with D > 2 dimensions, which was predicted using renormalization group arguments. Our finding should stimulate interest in the hierarchical Anderson model as a simplified and tractable model of the Anderson localization transition which occurs in finite-dimensional systems with short-range hopping.
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Submitted 22 April, 2013; v1 submitted 8 March, 2013;
originally announced March 2013.
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On the spectra of large sparse graphs with cycles
Authors:
D. Bollé,
F. L. Metz,
I. Neri
Abstract:
We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high accuracy. In particular, for (un)directed regular graphs with cycles of arbitrary length we derive exact and simple equations for the resolvent of the associated a…
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We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high accuracy. In particular, for (un)directed regular graphs with cycles of arbitrary length we derive exact and simple equations for the resolvent of the associated adjacency matrix. Solving these equations we obtain analytical formulas for the spectra and the boundaries of their support.
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Submitted 11 January, 2023; v1 submitted 7 June, 2012;
originally announced June 2012.
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Spectra of sparse non-Hermitian random matrices: an analytical solution
Authors:
I. Neri,
F. L. Metz
Abstract:
We present the exact analytical expression for the spectrum of a sparse non-Hermitian random matrix ensemble, generalizing two classical results in random-matrix theory: this analytical expression forms a non-Hermitian version of the Kesten-Mckay law as well as a sparse realization of Girko's elliptic law. Our exact result opens new perspectives in the study of several physical problems modelled o…
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We present the exact analytical expression for the spectrum of a sparse non-Hermitian random matrix ensemble, generalizing two classical results in random-matrix theory: this analytical expression forms a non-Hermitian version of the Kesten-Mckay law as well as a sparse realization of Girko's elliptic law. Our exact result opens new perspectives in the study of several physical problems modelled on sparse random graphs. In this context, we show analytically that the convergence rate of a transport process on a very sparse graph depends upon the degree of symmetry of the edges in a non-monotonous way.
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Submitted 23 December, 2012; v1 submitted 3 May, 2012;
originally announced May 2012.
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Spectra of sparse regular graphs with loops
Authors:
F. L. Metz,
I. Neri,
D. Bollé
Abstract:
We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization. Analytical formulas for the spectrum are obtain…
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We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization. Analytical formulas for the spectrum are obtained for specific length of the loops.
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Submitted 20 July, 2011;
originally announced July 2011.
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On the localization transition in symmetric random matrices
Authors:
F. L. Metz,
I. Neri,
D. Bollé
Abstract:
We study the behaviour of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully-connected Lévy matrices. We derive a critical line separating localized from extended states in the case of Lévy matrices. Comparison betw…
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We study the behaviour of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully-connected Lévy matrices. We derive a critical line separating localized from extended states in the case of Lévy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.
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Submitted 30 September, 2010; v1 submitted 20 May, 2010;
originally announced May 2010.
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The phase diagram of Lévy spin glasses
Authors:
I. Neri,
F. L. Metz,
D. Bollé
Abstract:
We study the Lévy spin-glass model with the replica and the cavity method. In this model each spin interacts through a finite number of strong bonds and an infinite number of weak bonds. This hybrid behaviour of Lévy spin glasses becomes transparent in our solution: the local field contains a part propagating along a backbone of strong bonds and a Gaussian noise term due to weak bonds. Our metho…
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We study the Lévy spin-glass model with the replica and the cavity method. In this model each spin interacts through a finite number of strong bonds and an infinite number of weak bonds. This hybrid behaviour of Lévy spin glasses becomes transparent in our solution: the local field contains a part propagating along a backbone of strong bonds and a Gaussian noise term due to weak bonds. Our method allows to determine the complete replica symmetric phase diagram, the replica symmetry breaking line and the entropy. The results are compared with simulations and previous calculations using a Gaussian ansatz for the distribution of fields.
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Submitted 7 October, 2009;
originally announced October 2009.
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Symmetric sequence processing in a recurrent neural network model with a synchronous dynamics
Authors:
F. L. Metz,
W. K. Theumann
Abstract:
The synchronous dynamics and the stationary states of a recurrent attractor neural network model with competing synapses between symmetric sequence processing and Hebbian pattern reconstruction is studied in this work allowing for the presence of a self-interaction for each unit. Phase diagrams of stationary states are obtained exhibiting phases of retrieval, symmetric and period-two cyclic stat…
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The synchronous dynamics and the stationary states of a recurrent attractor neural network model with competing synapses between symmetric sequence processing and Hebbian pattern reconstruction is studied in this work allowing for the presence of a self-interaction for each unit. Phase diagrams of stationary states are obtained exhibiting phases of retrieval, symmetric and period-two cyclic states as well as correlated and frozen-in states, in the absence of noise. The frozen-in states are destabilised by synaptic noise and well separated regions of correlated and cyclic states are obtained. Excitatory or inhibitory self-interactions yield enlarged phases of fixed-point or cyclic behaviour.
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Submitted 11 August, 2009;
originally announced August 2009.
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Instability of frozen-in states in synchronous Hebbian neural networks
Authors:
F. L. Metz,
W. K. Theumann
Abstract:
The full dynamics of a synchronous recurrent neural network model with Ising binary units and a Hebbian learning rule with a finite self-interaction is studied in order to determine the stability to synaptic and stochastic noise of frozen-in states that appear in the absence of both kinds of noise. Both, the numerical simulation procedure of Eissfeller and Opper and a new alternative procedure t…
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The full dynamics of a synchronous recurrent neural network model with Ising binary units and a Hebbian learning rule with a finite self-interaction is studied in order to determine the stability to synaptic and stochastic noise of frozen-in states that appear in the absence of both kinds of noise. Both, the numerical simulation procedure of Eissfeller and Opper and a new alternative procedure that allows to follow the dynamics over larger time scales have been used in this work. It is shown that synaptic noise destabilizes the frozen-in states and yields either retrieval or paramagnetic states for not too large stochastic noise. The indications are that the same results may follow in the absence of synaptic noise, for low stochastic noise.
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Submitted 13 May, 2008;
originally announced May 2008.
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Period-two cycles in a feed-forward layered neural network model with symmetric sequence processing
Authors:
F. L. Metz,
W. K. Theumann
Abstract:
The effects of dominant sequential interactions are investigated in an exactly solvable feed-forward layered neural network model of binary units and patterns near saturation in which the interaction consists of a Hebbian part and a symmetric sequential term. Phase diagrams of stationary states are obtained and a new phase of cyclic correlated states of period two is found for a weak Hebbian ter…
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The effects of dominant sequential interactions are investigated in an exactly solvable feed-forward layered neural network model of binary units and patterns near saturation in which the interaction consists of a Hebbian part and a symmetric sequential term. Phase diagrams of stationary states are obtained and a new phase of cyclic correlated states of period two is found for a weak Hebbian term, independently of the number of condensed patterns $c$.
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Submitted 19 April, 2007;
originally announced April 2007.
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Feed-forward chains of recurrent attractor neural networks with finite dilution near saturation
Authors:
F. L. Metz,
W. K. Theumann
Abstract:
A stationary state replica analysis for a dual neural network model that interpolates between a fully recurrent symmetric attractor network and a strictly feed-forward layered network, studied by Coolen and Viana, is extended in this work to account for finite dilution of the recurrent Hebbian interactions between binary Ising units within each layer. Gradual dilution is found to suppress part o…
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A stationary state replica analysis for a dual neural network model that interpolates between a fully recurrent symmetric attractor network and a strictly feed-forward layered network, studied by Coolen and Viana, is extended in this work to account for finite dilution of the recurrent Hebbian interactions between binary Ising units within each layer. Gradual dilution is found to suppress part of the phase transitions that arise from the competition between recurrent and feed-forward operation modes of the network. Despite that, a long chain of layers still exhibits a relatively good performance under finite dilution for a balanced ratio between inter-layer and intra-layer interactions.
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Submitted 9 November, 2005;
originally announced November 2005.
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Pattern reconstruction and sequence processing in feed-forward layered neural networks near saturation
Authors:
F. L. Metz,
W. K. Theumann
Abstract:
The dynamics and the stationary states for the competition between pattern reconstruction and asymmetric sequence processing are studied here in an exactly solvable feed-forward layered neural network model of binary units and patterns near saturation. Earlier work by Coolen and Sherrington on a parallel dynamics far from saturation is extended here to account for finite stochastic noise due to…
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The dynamics and the stationary states for the competition between pattern reconstruction and asymmetric sequence processing are studied here in an exactly solvable feed-forward layered neural network model of binary units and patterns near saturation. Earlier work by Coolen and Sherrington on a parallel dynamics far from saturation is extended here to account for finite stochastic noise due to a Hebbian and a sequential learning rule. Phase diagrams are obtained with stationary states and quasi-periodic non-stationary solutions. The relevant dependence of these diagrams and of the quasi-periodic solutions on the stochastic noise and on initial inputs for the overlaps is explicitly discussed.
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Submitted 1 July, 2005;
originally announced July 2005.