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Centrality metrics and localization in core-periphery networks

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2016年第2期000卷 023401-023401页

作者： Barucca, Paolo Tantari, Daniele Lillo, Fabrizio Scuola Normale Super Pisa Piazza Cavalieri 7 I-56126 Pisa PI Italy Ctr Ric Matemat Ennio De Giorgi Piazza Cavalieri 3 Pisa PI Italy

Two concepts of centrality have been defined in complex networks. The first considers the centrality of a node and many different metrics for it have been defined (e.g. eigenvector centrality, PageRank, non-backtracking centrality, etc). The second is related to large scale organization of the network, the core-periphery structure, composed by a dense core plus an outlying and loosely-connected periphery. In this paper we investigate the relation between these two concepts. We consider networks generated via the stochastic block model, or its degree corrected version, with a core-periphery structure and we investigate the centrality properties of the core nodes and the ability of several centrality metrics to identify them. We find that the three measures with the best performance are marginals obtained with belief propagation, PageRank, and degree centrality, while non-backtracking and eigenvector centrality (or MINRES [10], showed to be equivalent to the latter in the large network limit) perform worse in the investigated networks.

关键词： message-passing algorithms random graphs networks clustering techniques

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Estimator of prediction error based on approximate message passing for penalized linear regression

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2018年第6期2018卷 063404-063404页

作者： Sakata, Ayaka Inst Stat Math Midori Cho Tachikawa Tokyo 1908562 Japan Grad Univ Adv Sci SOKENDAI Hayama Kanagawa 2400193 Japan

We propose an estimator of prediction error using an approximate message passing (AMP) algorithm that can be applied to a broad range of sparse penalties. Following Stein's lemma, the estimator of the generalized degrees of freedom, which is a key quantity for the construction of the estimator of the prediction error, is calculated at the AMP fixed point. The resulting form of the AMP-based estimator does not depend on the penalty function, and its value can be further improved by considering the correlation between predictors. The proposed estimator is asymptotically unbiased when the components of the predictors and response variables are independently generated according to a Gaussian distribution. We examine the behavior of the estimator for real data under nonconvex sparse penalties, where Akaike's information criterion does not correspond to an unbiased estimator of the prediction error. The model selected by the proposed estimator is close to that which minimizes the true prediction error.

关键词： statistical inference learning theory message-passing algorithms

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Stochastic optimization by message passing

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2011年第11期2011卷

作者： Altarelli, F. Braunstein, A. Ramezanpour, A. Zecchina, R. Politecn Torino Dept Phys I-10129 Turin Italy Politecn Torino Ctr Computat Sci I-10129 Turin Italy Coll Carlo Alberto I-10024 Moncalieri Italy Human Genet Fdn I-10126 Turin Italy

Most optimization problems in applied sciences realistically involve uncertainty in the parameters defining the cost function, of which only statistical information is known beforehand. Here we provide an in-depth discussion of how message passing algorithms for stochastic optimization based on the cavity method of statistical physics can be constructed. We focus on two basic problems, namely the independent set problem and the matching problem, for which we display the general method and caveats for the case of the so called two-stage problem with independently distributed stochastic parameters. We compare the results with some greedy algorithms and briefly discuss the extension to more complicated stochastic multi-stage problems.

关键词： message-passing algorithms random graphs networks source and channel coding error correcting codes

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Circular coloring of random graphs: statistical physics investigation

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2016年第8期2016卷 083303-083303页

作者： Schmidt, Christian Guenther, Nils-Eric Zdeborova, Lenka CEA Saclay Inst Phys Theor F-91191 Gif Sur Yvette France CNRS F-91191 Gif Sur Yvette France Barcelona Inst Sci & Technol ICFO Inst Ciencies Foton Castelldefels 08860 Spain

Circular coloring is a constraint satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.

关键词： random graphs networks cavity and replica method inference of graphical models message-passing algorithms

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The cavity method to protein design problem

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2022年第10期2022卷 103403-103403页

作者： Takahashi, Tomoei Chikenji, George Tokita, Kei Nagoya Univ Grad Sch Informat Nagoya Aichi 4648601 Japan Nagoya Univ Grad Sch Engn Nagoya Aichi 4648603 Japan

In this study, we propose an analytic statistical mechanics approach to solve a fundamental problem in biological physics called protein design. Protein design is an inverse problem of protein structure prediction, and its solution is the amino acid sequence that best stabilizes a given conformation. Despite recent rapid progress in protein design using deep learning, the challenge of exploring protein design principles remains. Contrary to previous computational physics studies, we used the cavity method, an extension of the mean-field approximation that becomes rigorous when the interaction network is a tree. We found that for small two-dimensional lattice hydrophobic-polar protein models, the design by the cavity method yields results almost equivalent to those from the Markov chain Monte Carlo method with lower computational cost.

关键词： message-passing algorithms cavity and replica method protein function and design

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On one-step replica symmetry breaking in the Edwards-Anderson spin glass model

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2016年第7期2016卷

作者： Del Ferraro, Gino Wang, Chuang Zhou, Hai-Jun Aurell, Erik KTH Royal Inst Technol AlbaNova Univ Ctr Dept Computat Biol SE-10691 Stockholm Sweden Chinese Acad Sci Inst Theoret Phys State Key Lab Theoret Phys Beijing 100190 Peoples R China Harvard Univ Sch Engn & Appl Sci 33 Oxford St Cambridge MA 02138 USA KTH Royal Inst Technol ACCESS Linnaeus Ctr SE-10044 Stockholm Sweden KTH Royal Inst Technol Ctr Quantum Mat SE-10044 Stockholm Sweden Aalto Univ Dept Appl Phys FIN-00076 Aalto Finland Aalto Univ Dept Comp Sci FIN-00076 Aalto Finland

We consider a one-step replica symmetry breaking description of the Edwards-Anderson spin glass model in 2D. The ingredients of this description are a Kikuchi approximation to the free energy and a second-level statistical model built on the extremal points of the Kikuchi approximation, which are also fixed points of a generalized belief propagation (GBP) scheme. We show that a generalized free energy can be constructed where these extremal points are exponentially weighted by their Kikuchi free energy and a Parisi parameter y, and that the Kikuchi approximation of this generalized free energy leads to second-level, one-step replica symmetry breaking (1RSB), GBP equations. We then proceed analogously to the Bethe approximation case for tree-like graphs, where it has been shown that 1RSB belief propagation equations admit a survey propagation solution. We discuss when and how the one-step-replica symmetry breaking GBP equations that we obtain also allow a simpler class of solutions which can be interpreted as a class of generalized survey propagation equations for the single instance graph case.

关键词： message-passing algorithms optimization over networks spin glasses (theory) cavity and replica method

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Stability of the replica symmetric solution in diluted perceptron learning

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2013年第02期2013卷 L02002-L02002页

作者： Lage-Castellanos, Alejandro Pagnani, Andrea Quintero Angulo, Gretel Univ Havana Fac Phys Dept Theoret Phys Havana 10400 Cuba Univ Havana Fac Phys Henri Poincare Grp Complex Syst Havana Cuba Human Genet Fdn HuGeF Torino I-10122 Turin Italy

We study the role played by dilution in the average behavior of a perceptron model with continuous coupling with the replica method. We analyze the stability of the replica symmetric solution as a function of the dilution field for the generalization and memorization problems. Thanks to a Gardner-like stability analysis we show that at any fixed ratio a between the number of patterns M and the dimension N of the perceptron (alpha = M/N), there exists a critical dilution field h(c) above which the replica symmetric ansatz becomes unstable.

关键词： message-passing algorithms cognitive dynamical networks learning theory statistical inference

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Survey propagation as local equilibrium equations

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2004年第06期2004卷 P06007-P06007页

作者： Braunstein, A Zecchina, R SISSA I-34100 Trieste Italy Abdus Salaam Int Ctr Theoret Phys I-34100 Trieste Italy

It has been shown experimentally that a decimation algorithm based on survey propagation (SP) equations allows one to solve efficiently some combinatorial problems over random graphs. We show that these equations can be derived as sum - product equations for the computation of marginals in an extended space where the variables are allowed to take an additional value - - when they are not forced by the combinatorial constraints. An appropriate 'local equilibrium condition' cost/energy function is introduced and its entropy is shown to coincide with the expected logarithm of the number of clusters of solutions as computed by SP. These results may help to clarify the geometrical notion of clusters assumed by SP for random K-SAT or random graph colouring ( where it is conjectured to be exact) and help to explain which kind of clustering operation or approximation is enforced in general/small sized models in which it is known to be inexact.

关键词： cavity and replica method message-passing algorithms source and channel coding

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Characterizing and improving generalized belief propagation algorithms on the 2D Edwards-Anderson model

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2011年第12期2011卷 P12007-P12007页

作者： Dominguez, Eduardo Lage-Castellanos, Alejandro Mulet, Roberto Ricci-Tersenghi, Federico Rizzo, Tommaso Univ Havana Dept Theoret Phys Havana Cuba Univ Havana Fac Phys Henri Poincare Grp Complex Syst Havana Cuba Univ Roma La Sapienza Dipartimento Fis Ist Nazl Fis Nucl Sez Roma 1 I-00185 Rome Italy Univ Roma La Sapienza CNR IPCF UOS Roma I-00185 Rome Italy

We study the performance of different message passing algorithms in the two-dimensional Edwards-Anderson model. We show that the standard belief propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a generalized belief propagation (GBP) algorithm, derived from a cluster variational method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: double loop (DL) and a two-way message passing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP message passing algorithm converges also to nonparamagnetic solutions. The lack of convergence of the standard GBP message passing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new message passing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude.

关键词： cavity and replica method disordered systems (theory) message-passing algorithms statistical inference

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The number of matchings in random graphs

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JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT 2006年第05期2006卷 P05003-P05003页

作者： Zdeborova, Lenka Mezard, Marc Univ Paris 11 CNRS UMR 8626 LPTMS F-91405 Orsay France

We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.

关键词： cavity and replica method message-passing algorithms random graphs networks

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