Cavity Method Research Articles

Bad Local Minima Exist in the Stochastic Block Model

AbstractWe study the disassortative stochastic block model with three communities, a well-studied model of graph partitioning and Bayesian inference for which detailed predictions based on the cavity method exist (Decelle et al. in Phys Rev E 84:066106, 2011). We provide strong evidence that for a part of the phase where efficient algorithms exist that approximately reconstruct the communities, inference based on maximum a posteriori (MAP) fails. In other words, we show that there exist modes of the posterior distribution that have a vanishing agreement with the ground truth. The proof is based on the analysis of a graph colouring algorithm from Achlioptas and Moore (J Comput Syst Sci 67:441–471, 2003).

Phase transition analysis of the Potts-SOS model with spin set {−1,0,+1} on the Cayley tree

Abstract Recently, the Potts-SOS model on Cayley trees was studied using the Kolmogorov consistency condition to investigate the existence and properties of Gibbs measures. The author previously examined the thermodynamic properties of the one-dimensional Potts-SOS model on the positive natural number lattice N , demonstrating the absence of phase transitions [Akin H, Phys. Scr. 99 (2024), 055 231]. In this study, we apply the cavity method to explore the non-uniqueness of Gibbs measures for the Potts-SOS model with spins {–1, 0, + 1} on a second-order semi-infinite Cayley tree. We address the phase transition problem by analyzing the model’s iterative equation system and perform stability analysis at fixed points to determine regions where phase transitions occur.

Sparse representations, inference and learning

Abstract In recent years statistical physics has proven to be a valuable tool to probe into large dimensional inference problems such as the ones occurring in machine learning. Statistical physics provides analytical tools to study fundamental limitations in their solutions and proposes algorithms to solve individual instances. In these notes, based on the lectures by Marc Mézard in 2022 at the summer school in Les Houches, we will present a general framework that can be used in a large variety of problems with weak long-range interactions, including the compressed sensing problem, or the problem of learning in a perceptron. We shall see how these problems can be studied at the replica symmetric level, using developments of the cavity methods, both as a theoretical tool and as an algorithm.

Mechanical behaviour of rigid-flexible combined structures: Aluminium-inflated membrane beams for application in floating photovoltaic platform

This study aims to investigate the bending and failure behaviour of aluminium-inflated membrane beams for their applications in floating photovoltaic platforms. Four-point bending tests are conducted for a range of inflated pressures and two different deck heights to assess their effect on structural stiffness and ultimate bearing capacity. Meanwhile, the surface-based fluid cavity method is employed to develop the finite element model with the material properties determined by independent coupon-level tests. The bearing capacity of the aluminium-inflated membrane beam is positively correlated with the internal pressure and deck height. The midspan strain distribution is similar to those of the traditional four-point bending beam with the upper part undergoing compression and the lower part experiencing tension, however, the structural behaviour at the failure stage is different. Failure typically occurs due to localised depressions at the loading points on the aluminium deck, ultimately leading to structural failure. The numerical model closely matches the experimental data for the initial inflated and bending configurations, exhibiting a deviation of only 0.10 % to 0.46 % in diameter and 0.32 % to 5.57 % in equivalent bending stiffness. A parametric study shows that the loading properties of the beam are more sensitive to the internal pressure than the deck height and thickness.

Counting and hardness-of-finding fixed points in cellular automata on random graphs

Abstract We study the fixed points of outer-totalistic cellular automata on sparse random regular graphs. These can be seen as constraint satisfaction problems, where each variable must adhere to the same local constraint, which depends solely on its state and the total number of its neighbors in each possible state. Examples of this setting include classical problems such as independent sets or assortative/dissasortative partitions. We analyse the existence and number of fixed points in the large system limit using the cavity method, under both the replica symmetric (RS) and one-step replica symmetry breaking (1RSB) assumption. This method allows us to characterize the structure of the space of solutions, in particular, if the solutions are clustered and whether the clusters contain frozen variables. This last property is conjectured to be linked to the typical algorithmic hardness of the problem. We bring experimental evidence for this claim by studying the performance of the belief-propagation reinforcement algorithm, a message-passing-based solver for these constraint satisfaction problems.

All-solid continuous-wave Nd: YVO4-diamond intracavity Raman laser

We demonstrated a continuous intracavity diamond Raman laser operating by linear-cavity and V-cavity design. The diamond crystal as the Raman medium and Nd: YVO4 as the laser gain medium was utilized to achieve high beam quality Stokes output with narrow line width. By employing linear and V-shaped cavity, we achieved Stokes light outputs of 1.68 and 2.26 W, respectively. Compared to the linear cavity, the V-shaped cavity has a higher conversion efficiency of 11.07% and a narrower linewidth of 0.07 nm. Mechanism for linewidth of different cavity design and methods for further increasing power range and high quality are discussed.

Geometric Landscape Annealing as an Optimization Principle Underlying the Coherent Ising Machine

Given the fundamental importance of combinatorial optimization across many diverse domains, there has been widespread interest in the development of unconventional physical computing architectures that can deliver better solutions with lower resource costs. However, a theoretical understanding of their performance remains elusive. We develop such understanding for the case of the coherent Ising machine (CIM), a network of optical parametric oscillators that can be applied to any quadratic unconstrained binary optimization problem. We focus on how the CIM finds low-energy solutions of the Sherrington-Kirkpatrick spin glass. As the laser gain of this system is annealed, the CIM interpolates between gradient descent on coupled soft spins to descent on coupled binary spins. By combining the Kac-Rice formula, the replica method, and supersymmetry breaking, we develop a detailed understanding of the evolving geometry of the high-dimensional energy landscape of the CIM as the laser gain increases, finding several phase transitions in the landscape, from flat to rough to rigid. Additionally, we develop a novel cavity method that provides a geometric interpretation of supersymmetry breaking in terms of the reactivity of a rough landscape to specific external perturbations. Our energy landscape theory successfully matches numerical experiments, provides geometric insights into the principles of CIM operation, and yields optimal annealing schedules. Published by the American Physical Society 2024

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node's importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its "global" character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability [Formula: see text] that a random node in the ensemble of undirected random graphs has centrality [Formula: see text] satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank-[Formula: see text] projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

Resonant cavity method based on RF measurement and simulation for measuring complex permittivity or permeability of RF absorbing materials

Accurate measurement of complex permittivity and/or permeability of radio-frequency (RF) absorbing materials is essential to predict the performances of higher-order-mode damping in damped accelerating cavities or in vacuum chambers. We propose an improved resonant cavity method that can be used as a complement to the widely used Nicolson–Ross–Weir method for measuring the RF characteristics of materials. The conventional resonant cavity method is based on the cavity perturbation theory. However, the measurements are inaccurate when losses in the materials are large. The improved method we propose determines the complex permittivity or permeability of materials from the measured resonant frequencies and Q factors based on eigenmode simulations. Because the reliability on the eigenmode calculation for RF cavities with lossy materials has become sufficiently reliable, determination of complex permittivity and permeability can be achieved using such a simulation software. We demonstrate this method using ferrite HF70 (TDK Corp.), which is useful for RF absorbers in damped accelerating cavities or in vacuum chambers.

Nature of the Volcano Transition in the Fully Disordered Kuramoto Model.

Randomly coupled phase oscillators may synchronize into disordered patterns of collective motion. We analyze this transition in a large, fully connected Kuramoto model with symmetric but otherwise independent random interactions. Using the dynamical cavity method, we reduce the dynamics to a stochastic single-oscillator problem with self-consistent correlation and response functions that we study analytically and numerically. We clarify the nature of the volcano transition and elucidate its relation to the existence of an oscillator glass phase.

Dielectric response characteristics and moisture exposure evaluation of oil-paper insulation based on microstrip ring resonator

Moisture ingress significantly accelerates the aging of oil-paper insulated equipment, making accurate assessment of moisture content critical for the operation and maintenance of oil-immersed insulated power equipment. In this paper, a novel method for rapid moisture content evaluation is proposed based on a microstrip ring resonator (MRR) designed with a fundamental frequency of 1.16 GHz. A calculation model for the dielectric response analysis using MRR is established to characterize dielectric properties of the material layered on the substrate. ANSYS HFSS electromagnetic simulation software was used to study the impact of dielectric material thickness and the substrate on the MRR test results. Experimental measurement demonstrated that the maximum deviation in resonance frequency was merely 0.542 %. The dielectric properties of uniformly and non-uniformly damp insulating pressboard were analyzed across the frequency band from 1 to 6 GHz. The MRR test results were compared with those of the standard resonant cavity method and frequency domain dielectric spectroscopy (FDS) test. Using the regression algorithm and curve analysis, moisture assessment models for both uniformly and non-uniformly damp oil-paper insulation were established. The experimental results show that the MRR test can clearly distinguish between the pressboard samples with varying moisture levels under uniform conditions, with a maximum moisture evaluation error of 8.19 %. The results demonstrate that the MRR test method, requiring only 56 s, can quickly and effectively evaluate the moisture content of insulating pressboard.

Dynamical phase transitions in graph cellular automata.

Discrete dynamical systems can exhibit complex behavior from the iterative application of straightforward local rules. A famous class of examples comes from cellular automata whose global dynamics are notoriously challenging to analyze. To address this, we relax the regular connectivity grid of cellular automata to a random graph, which gives the class of graph cellular automata. Using the dynamical cavity method and its backtracking version, we show that this relaxation allows us to derive asymptotically exact analytical results on the global dynamics of these systems on sparse random graphs. Concretely, we showcase the results on a specific subclass of graph cellular automata with "conforming nonconformist" update rules, which exhibit dynamics akin to opinion formation. Such rules update a node's state according to the majority within their own neighborhood. In cases where the majority leads only by a small margin over the minority, nodes may exhibit nonconformist behavior. Instead of following the majority, they either maintain their own state, switch it, or follow the minority. For configurations with different initial biases towards one state we identify sharp dynamical phase transitions in terms of the convergence speed and attractor types. From the perspective of opinion dynamics this answers when consensus will emerge and when two opinions coexist almost indefinitely.

Deflation analysis of an air cushion for underwater shaking tables

The deflation process of inflatable membrane structures has always been a topic worthy of in-depth research. Inflatable membrane structures are employed for waterproofing in underwater environments, and their deflation process often poses safety concerns, warranting increased attention. An air cushion, which is applied to waterproof the underwater shaking table, is researched. It is arranged between the shaking table and the cover plates and can provide waterproof protection for the driving equipment of the shaking table. This article mainly studies the complete process of deflation for the waterproof air cushion. Force analysis was conducted on the air cushion membrane at different stages of deflation, and the factors that affect force during the deflation process were determined. A simulation method that combines the CEL (Coupled Eulerian-Lagrangian) method and fluid cavity method was used to analyze the deflation process of the air cushion, taking into account the effects of different deflation speeds and initial inflation pressures on the process. Based on the analysis results of the pressure-air volume curve and the deformation of the air cushion, the deflation process can be divided into three stages: elastic shrinkage stage, extrusion deformation stage, and free deformation stage. Based on the analysis results of dynamic stability, the extrusion deformation stage is further divided into two sub-stages: the extrusion stable stage and the extrusion unstable stage. It was found that the air cushion can achieve stability during both the elastic shrinkage stage and the extrusion stable stage, leading to the proposal of an evaluation method for air cushion deflation. Finally, the instantaneous leakage process of different positions on the air cushion was analyzed, and engineering suggestions were provided based on the analysis results. The research results provide a reference for the application of underwater inflatable membrane structures.

Effect of constraint relaxation on the minimum vertex cover problem in random graphs.

A statistical-mechanical study of the effect of constraint relaxation on the minimum vertex cover problem in Erdős-Rényi random graphs is presented. Using a penalty-method formulation for constraint relaxation, typical properties of solutions, including infeasible solutions that violate the constraints, are analyzed by means of the replica method and cavity method. The problem involves a competition between reducing the number of vertices to be covered and satisfying the edge constraints. The analysis under the replica-symmetric (RS) ansatz clarifies that the competition leads to degeneracies in the vertex and edge states, which determine the quantitative properties of the system, such as the cover and penalty ratios. A precise analysis of these effects improves the accuracy of RS approximation for the minimum cover ratio in the replica symmetry-breaking (RSB) region. Furthermore, the analysis based on the RS cavity method indicates that the RS/RSB boundary of the ground states with respect to the mean degree of the graphs is expanded, and the critical temperature is lowered by constraint relaxation.

Understanding microwave heating of oils

Our recent study found that vegetable oils, often deemed unsuitable for microwave heating, actually heat efficiently in microwave ovens. To explore the main chemical components within vegetable oil that contribute to microwave heating, we considered two hypotheses: 1) the presence of water dissolved in oil and 2) the glycerol component of oil molecule. In this research, we used the resonant cavity method to measure dielectric properties of vegetable oil with different water activities and compared vegetable oil with mineral oil (which does not contain the glycerol component). Our results revealed no significant difference in the dielectric properties or microwave heating rates between vegetable oils with low (0.16) and high (1.00) water activities, indicating that dissolved water is not a major contributor. Regardless of the water activity levels, vegetable oil heats well in microwaves, whereas mineral oil does not. The microwave heating difference was elucidated by the measured dielectric properties: the loss factor of vegetable oil (0.15) was 30 times greater than that of mineral oil (0.005). This supports our hypothesis that the glycerol component in vegetable oil contributes to microwave heating, whereas its absence in mineral oil accounts for the limited microwave heating capability. This research augments our fundamental comprehension of microwave heating of oils.

Cavity approach for the approximation of spectral density of graphs with heterogeneous structures.

Graphs have become widely used to represent and study social, biological, and technological systems. Statistical methods to analyze empirical graphs were proposed based on the graph's spectral density. However, their running time is cubic in the number of vertices, precluding direct application to large instances. Thus, efficient algorithms to calculate the spectral density become necessary. For sparse graphs, the cavity method can efficiently approximate the spectral density of locally treelike undirected and directed graphs. However, it does not apply to most empirical graphs because they have heterogeneous structures. Thus, we propose methods for undirected and directed graphs with heterogeneous structures using a new vertex's neighborhood definition and the cavity approach. Our methods' time and space complexities are O(|E|h_{max}^{3}t) and O(|E|h_{max}^{2}t), respectively, where |E| is the number of edges, h_{max} is the size of the largest local neighborhood of a vertex, and t is the number of iterations required for convergence. We demonstrate the practical efficacy by estimating the spectral density of simulated and real-world undirected and directed graphs.

Introduction to dynamical mean-field theory of randomly connected neural networks with bidirectionally correlated couplings

Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.

Emergent competition shapes top-down versus bottom-up control in multi-trophic ecosystems.

Ecosystems are commonly organized into trophic levels-organisms that occupy the same level in a food chain (e.g., plants, herbivores, carnivores). A fundamental question in theoretical ecology is how the interplay between trophic structure, diversity, and competition shapes the properties of ecosystems. To address this problem, we analyze a generalized Consumer Resource Model with three trophic levels using the zero-temperature cavity method and numerical simulations. We derive the corresponding mean-field cavity equations and show that intra-trophic diversity gives rise to an effective "emergent competition" term between species within a trophic level due to feedbacks mediated by other trophic levels. This emergent competition gives rise to a crossover from a regime of top-down control (populations are limited by predators) to a regime of bottom-up control (populations are limited by primary producers) and is captured by a simple order parameter related to the ratio of surviving species in different trophic levels. We show that our theoretical results agree with empirical observations, suggesting that the theoretical approach outlined here can be used to understand complex ecosystems with multiple trophic levels.

Design of the electrical system on a general cargo ship with a length of 105,669 meters

This research investigates the incorporation of LED lights and the zonal cavity method in the initial design stage of a cargo-type ship, focusing on efficient lighting for diverse ship zones. The study aims to determine optimal lighting requirements, adhering to industry standards, for zones such as double bottom, maindeck, poopdeck, boatdeck, bridgedeck, navdeck, and emergency conditions. Ships play a vital role in transporting passengers and goods across seas and rivers, necessitating effective lighting systems for safe operations, particularly during nighttime. This paper explores the integration of LED lights and the zonal cavity method in the general arrangement design of a cargo ship. The zonal cavity method is applied to ascertain lighting intensity in different ship zones based on room dimensions, including width, height, length, and area. Illumination values are determined according to established standards for each zone. The cumulative lighting electricity load for various ship zones using LED lights is calculated to identify specific power requirements.The lighting power requirements for the double bottom, maindeck, poopdeck, boatdeck, bridgedeck, navdeck, and emergency conditions are found to be 0.141 KW, 0.75 KW, 1.068 KW, 0.762 KW, 0.671 KW, 0.42 KW, and 0.296 KW, respectively. The total lighting power for all zones is calculated as 3.7776 KW, suggesting the installation of a 4 KW generator for optimal power supply.The application of the zonal cavity method in the planning and design of a cargo ship's lighting system, combined with energy-efficient LED lights, proves to be an effective approach. The results indicate that a 4 KW generator is sufficient to meet the lighting power demands for various ship zones, ensuring compliance with safety standards and technological advancements in maritime lighting.

Statistical mechanics of inference in epidemic spreading.

We investigate the information-theoretical limits of inference tasks in epidemic spreading on graphs in the thermodynamic limit. The typical inference tasks consist in computing observables of the posterior distribution of the epidemic model given observations taken from a ground-truth (sometimes called planted) random trajectory. We can identify two main sources of quenched disorder: the graph ensemble and the planted trajectory. The epidemic dynamics however induces nontrivial long-range correlations among individuals' states on the latter. This results in nonlocal correlated quenched disorder which unfortunately is typically hard to handle. To overcome this difficulty, we divide the dynamical process into two sets of variables: a set of stochastic independent variables (representing transmission delays), plus a set of correlated variables (the infection times) that depend deterministically on the first. Treating the former as quenched variables and the latter as dynamic ones, computing disorder average becomes feasible by means of the replica-symmetric cavity method. We give theoretical predictions on the posterior probability distribution of the trajectory of each individual, conditioned to observations on the state of individuals at given times, focusing on the susceptible infectious (SI) model. In the Bayes-optimal condition, i.e., when true dynamic parameters are known, the inference task is expected to fall in the replica-symmetric regime. We indeed provide predictions for the information theoretic limits of various inference tasks, in form of phase diagrams. We also identify a region, in the Bayes-optimal setting, with strong hints of replica-symmetry breaking. When true parameters are unknown, we show how a maximum-likelihood procedure is able to recover them with mostly unaffected performance.