1
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Das S. Multiple transitions in an infinite range p-spin random crystal field Blume-Capel model. Phys Rev E 2022; 106:034134. [PMID: 36266859 DOI: 10.1103/physreve.106.034134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2022] [Accepted: 09/05/2022] [Indexed: 06/16/2023]
Abstract
We study a p-spin model with ferromagnetic coupling and quenched random crystal fields for p≥3 for spin-1 systems. We find that the model has lines of first-order transitions at finite temperature (T) for all p≥3. For bimodal distribution of the random crystal field these lines meet at a triple point for weak strength of the crystal field (Δ). Beyond a critical strength of Δ, they do not meet and one of the lines ends at a critical point (T_{c}). Interestingly, we find that on increasing T from T_{c}, keeping other parameters fixed, the system undergoes one more transition which is first order in its character. The system thus exhibits a Gardner-like transition for a range of parameters for all finite p≥3. For p→∞ the model behaves differently and there is only one random first-order transition at T=0.
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Affiliation(s)
- Santanu Das
- School of Physical Sciences, National Institute of Science Education and Research, Jatni 752050, India and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar 400094, India
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2
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Torrisi G, Annibale A, Kühn R. Overcoming the complexity barrier of the dynamic message-passing method in networks with fat-tailed degree distributions. Phys Rev E 2021; 104:045313. [PMID: 34781444 DOI: 10.1103/physreve.104.045313] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2021] [Accepted: 10/10/2021] [Indexed: 11/07/2022]
Abstract
The dynamic cavity method provides the most efficient way to evaluate probabilities of dynamic trajectories in systems of stochastic units with unidirectional sparse interactions. It is closely related to sum-product algorithms widely used to compute marginal functions from complicated global functions of many variables, with applications in disordered systems, combinatorial optimization, and computer science. However, the complexity of the cavity approach grows exponentially with the in-degrees of the interacting units, which creates a defacto barrier for the successful analysis of systems with fat-tailed in-degree distributions. In this paper, we present a dynamic programming algorithm that overcomes this barrier by reducing the computational complexity in the in-degrees from exponential to quadratic, whenever couplings are chosen randomly from (or can be approximated in terms of) discrete, possibly unit-dependent, sets of equidistant values. As a case study, we analyze the dynamics of a random Boolean network with a fat-tailed degree distribution and fully asymmetric binary ±J couplings, and we use the power of the algorithm to unlock the noise-dependent heterogeneity of stationary node activation patterns in such a system.
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Affiliation(s)
- Giuseppe Torrisi
- Department of Mathematics, King's College London, Strand, and London WC2R 2LS, United Kingdom
| | - Alessia Annibale
- Department of Mathematics, King's College London, Strand, and London WC2R 2LS, United Kingdom
| | - Reimer Kühn
- Department of Mathematics, King's College London, Strand, and London WC2R 2LS, United Kingdom
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3
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Choi JM, Gilson AI, Shakhnovich EI. Graph's Topology and Free Energy of a Spin Model on the Graph. PHYSICAL REVIEW LETTERS 2017; 118:088302. [PMID: 28282198 PMCID: PMC5668130 DOI: 10.1103/physrevlett.118.088302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2016] [Indexed: 06/06/2023]
Abstract
In this Letter we investigate a direct relationship between a graph's topology and the free energy of a spin system on the graph. We develop a method of separating topological and energetic contributions to the free energy, and find that considering the topology is sufficient to qualitatively compare the free energies of different graph systems at high temperature, even when the energetics are not fully known. This method was applied to the metal lattice system with defects, and we found that it partially explains why point defects are more stable than high-dimensional defects. Given the energetics, we can even quantitatively compare free energies of different graph structures via a closed form of linear graph contributions. The closed form is applied to predict the sequence-space free energy of lattice proteins, which is a key factor determining the designability of a protein structure.
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Affiliation(s)
- Jeong-Mo Choi
- Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, USA
| | - Amy I Gilson
- Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, USA
| | - Eugene I Shakhnovich
- Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, Massachusetts 02138, USA
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4
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Qin SM, Zeng Y, Zhou HJ. Spin-glass phase transitions and minimum energy of the random feedback vertex set problem. Phys Rev E 2016; 94:022146. [PMID: 27627285 DOI: 10.1103/physreve.94.022146] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2016] [Indexed: 11/07/2022]
Abstract
A feedback vertex set (FVS) of an undirected graph contains vertices from every cycle of this graph. Constructing a FVS of sufficiently small cardinality is very difficult in the worst cases, but for random graphs this problem can be efficiently solved by converting it into an appropriate spin-glass model [H.-J. Zhou, Eur. Phys. J. B 86, 455 (2013)EPJBFY1434-602810.1140/epjb/e2013-40690-1]. In the present work we study the spin-glass phase transitions and the minimum energy density of the random FVS problem by the first-step replica-symmetry-breaking (1RSB) mean-field theory. For both regular random graphs and Erdös-Rényi graphs, we determine the inverse temperature β_{l} at which the replica-symmetric mean-field theory loses its local stability, the inverse temperature β_{d} of the dynamical (clustering) phase transition, and the inverse temperature β_{s} of the static (condensation) phase transition. These critical inverse temperatures all change with the mean vertex degree in a nonmonotonic way, and β_{d} is distinct from β_{s} for regular random graphs of vertex degrees K>60, while β_{d} are identical to β_{s} for Erdös-Rényi graphs at least up to mean vertex degree c=512. We then derive the zero-temperature limit of the 1RSB theory and use it to compute the minimum FVS cardinality.
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Affiliation(s)
- Shao-Meng Qin
- Institute of Theoretical Physics, Chinese Academy of Sciences, Zhong-Guan-Cun East Road 55, Beijing 100190, China.,College of Science, Civil Aviation University of China, Tianjin 300300, China
| | - Ying Zeng
- Beijing National Laboratory for Condensed Matter Physics and Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Hai-Jun Zhou
- Institute of Theoretical Physics, Chinese Academy of Sciences, Zhong-Guan-Cun East Road 55, Beijing 100190, China.,School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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5
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Del Ferraro G, Aurell E. Dynamic message-passing approach for kinetic spin models with reversible dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:010102. [PMID: 26274101 DOI: 10.1103/physreve.92.010102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2014] [Indexed: 06/04/2023]
Abstract
A method to approximately close the dynamic cavity equations for synchronous reversible dynamics on a locally treelike topology is presented. The method builds on (a) a graph expansion to eliminate loops from the normalizations of each step in the dynamics and (b) an assumption that a set of auxilary probability distributions on histories of pairs of spins mainly have dependencies that are local in time. The closure is then effectuated by projecting these probability distributions on n-step Markov processes. The method is shown in detail on the level of ordinary Markov processes (n=1) and outlined for higher-order approximations (n>1). Numerical validations of the technique are provided for the reconstruction of the transient and equilibrium dynamics of the kinetic Ising model on a random graph with arbitrary connectivity symmetry.
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Affiliation(s)
- Gino Del Ferraro
- Department of Computational Biology, AlbaNova University Centre, SE-106 91 Stockholm, Sweden
| | - Erik Aurell
- Department of Computational Biology, AlbaNova University Centre, SE-106 91 Stockholm, Sweden
- ACCESS Linnaeus Centre, KTH-Royal Institute of Technology, SE-100 44 Stockholm, Sweden
- Departments of Information and Computer Science and Applied Physics and Aalto Science Institute, Aalto University, P. O. Box 15400, FI-00076 Aalto, Finland
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6
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Takabe S, Hukushima K. Minimum vertex cover problems on random hypergraphs: replica symmetric solution and a leaf removal algorithm. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062139. [PMID: 25019756 DOI: 10.1103/physreve.89.062139] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2013] [Indexed: 06/03/2023]
Abstract
The minimum vertex-cover problems on random α-uniform hypergraphs are studied using two different approaches, a replica method in statistical mechanics of random systems and a leaf removal algorithm. It is found that there exists a phase transition at the critical average degree e/(α-1), below which a replica symmetric ansatz in the replica method holds and the algorithm estimates exactly the same solution of the problem as that by the replica method. In contrast, above the critical degree, the replica symmetric solution becomes unstable and the leaf-removal algorithm fails to estimate the optimal solution because of the emergence of a large size core. These results strongly suggest a close relation between the replica symmetry and the performance of an approximation algorithm. Critical properties of the core percolation are also examined numerically by a finite-size scaling.
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Affiliation(s)
- Satoshi Takabe
- Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
| | - Koji Hukushima
- Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
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7
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Jurčišinová E, Jurčišin M. Phase transitions of the p-spin model on pure Husimi lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:012140. [PMID: 23944447 DOI: 10.1103/physreve.88.012140] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/05/2013] [Revised: 06/15/2013] [Indexed: 06/02/2023]
Abstract
We consider the p-spin model with spin 1/2 on all pure Husimi lattices. Using an effective representation of the recursion relations, the phase transitions of the model on all pure Husimi lattices are investigated. First, the nonexistence of the second order phase transitions in the model on all pure Husimi lattices is proven exactly. Then the existence and properties of the first order phase transitions in a zero external magnetic field are studied in detail. An implicit polynomial equation for determining temperatures below which the paramagnetic and ferromagnetic phases coexist in the model on all pure Husimi lattices is found. In addition, an implicit equation for exactly determining the transition temperatures of the first order phase transitions in a zero external magnetic field on all pure Husimi lattices is derived and discussed.
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Affiliation(s)
- E Jurčišinová
- Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01 Košice, Slovakia
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8
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Ferreira AL, Mendes JFF, Ostilli M. First- and second-order phase transitions in Ising models on small-world networks: simulations and comparison with an effective field theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:011141. [PMID: 20866599 DOI: 10.1103/physreve.82.011141] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2010] [Revised: 05/27/2010] [Indexed: 05/29/2023]
Abstract
We perform simulations of random Ising models defined over small-world networks and we check the validity and the level of approximation of a recently proposed effective field theory. Simulations confirm a rich scenario with the presence of multicritical points with first- or second-order phase transitions. In particular, for second-order phase transitions, independent of the dimension d0 of the underlying lattice, the exact predictions of the theory in the paramagnetic regions, such as the location of critical surfaces and correlation functions, are verified. Quite interestingly, we verify that the Edwards-Anderson model with d0=2 is not thermodynamically stable under graph noise.
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Affiliation(s)
- A L Ferreira
- Departamento de Física and I3N, Universidade de Aveiro, 3810-193 Aveiro, Portugal
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9
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Jörg T, Krzakala F, Semerjian G, Zamponi F. First-order transitions and the performance of quantum algorithms in random optimization problems. PHYSICAL REVIEW LETTERS 2010; 104:207206. [PMID: 20867059 DOI: 10.1103/physrevlett.104.207206] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/19/2009] [Revised: 03/30/2010] [Indexed: 05/26/2023]
Abstract
We present a study of the phase diagram of a random optimization problem in the presence of quantum fluctuations. Our main result is the characterization of the nature of the phase transition, which we find to be a first-order quantum phase transition. We provide evidence that the gap vanishes exponentially with the system size at the transition. This indicates that the quantum adiabatic algorithm requires a time growing exponentially with system size to find the ground state of this problem.
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Affiliation(s)
- Thomas Jörg
- LPTENS, CNRS UMR 8549, associée à l'UPMC Paris 06, 24 Rue Lhomond, 75005 Paris, France
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10
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Ostilli M, Mendes JFF. Communication and correlation among communities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011142. [PMID: 19658688 DOI: 10.1103/physreve.80.011142] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2009] [Indexed: 05/28/2023]
Abstract
Given a network and a partition in communities, we consider the issues "how communities influence each other" and "when two given communities do communicate." Specifically, we address these questions in the context of small-world networks, where an arbitrary quenched graph is given and long-range connections are randomly added. We prove that, among the communities, a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented by M. Ostilli and J. F. F. Mendes [Phys. Rev. E 78, 031102 (2008)] (n=1), which here (n>1) consists in a sort of effective TAP (Thouless, Anderson, and Palmer) equations in which each community plays the role of a microscopic spin. The relative susceptibilities derived from these equations calculated at finite or zero temperature, where the method provides an effective percolation theory, give us the answers to the above issues. Unlike the case n=1, asymmetries among the communities may lead, via the TAP-like structure of the equations, to many metastable states whose number, in the case of negative shortcuts among the communities, may grow exponentially fast with n. As examples we consider the n Viana-Bray communities model and the n one-dimensional small-world communities model. Despite being the simplest ones, the relevance of these models in network theory, as, e.g., in social networks, is crucial and no analytic solution were known until now. Connections between percolation and the fractal dimension of a network are also discussed. Finally, as an inverse problem, we show how, from the relative susceptibilities, a natural and efficient method to detect the community structure of a generic network arises.
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Affiliation(s)
- M Ostilli
- Departamento de Física, Universidade de Aveiro, 3810-193 Aveiro, Portugal
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11
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Nakajima T, Hukushima K. Thermodynamic construction of a one-step replica-symmetry-breaking solution in finite-connectivity spin glasses. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011103. [PMID: 19658649 DOI: 10.1103/physreve.80.011103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2009] [Revised: 05/07/2009] [Indexed: 05/28/2023]
Abstract
A one-step replica-symmetry-breaking solution for finite-connectivity spin-glass models with K body interaction is constructed at finite temperature using the replica method and thermodynamic constraints. In the absence of external fields, this construction provides a general extension of replica symmetric solution at finite replica number to one-step replica-symmetry-breaking solution. It is found that this result is formally equivalent to that of the one-step replica-symmetry-breaking cavity method. To confirm the validity of the obtained solution, Monte Carlo simulations are performed for K=2 and 3. The thermodynamic quantities of the Monte Carlo results extrapolated to a large-size limit are consistent with those estimated by our solution for K=2 at all simulated temperatures and for K=3 except near the transition temperature.
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Affiliation(s)
- T Nakajima
- Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan.
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12
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Alamino RC, Saad D. Typical behavior of relays in communication channels. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:061124. [PMID: 18233831 DOI: 10.1103/physreve.76.061124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2007] [Revised: 10/26/2007] [Indexed: 05/25/2023]
Abstract
The typical behavior of the relay-without-delay channel under low-density parity-check coding and its multiple-unit generalization, termed the relay array, is studied using methods of statistical mechanics. A demodulate-and-forward strategy is analytically solved using the replica symmetric ansatz which is exact in the system studied at Nishimori's temperature. In particular, the typical level of improvement in communication performance by relaying messages is shown in the case of a small and a large number of relay units.
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Affiliation(s)
- R C Alamino
- Neural Computing Research Group, Aston University, Birmingham, United Kingdom
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13
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Migliorini G, Saad D. Finite-connectivity spin-glass phase diagrams and low-density parity check codes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:026122. [PMID: 16605413 DOI: 10.1103/physreve.73.026122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2005] [Indexed: 05/08/2023]
Abstract
We obtain phase diagrams of regular and irregular finite-connectivity spin glasses. Contact is first established between properties of the phase diagram and the performance of low-density parity check (LDPC) codes within the replica symmetric (RS) ansatz. We then study the location of the dynamical and critical transition points of these systems within the one step replica symmetry breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that the location of the dynamical transition line does change within the RSB theory, in comparison with the results obtained in the RS case. For LDPC decoding of messages transmitted over the binary erasure channel we find, at zero temperature and rate , an RS critical transition point at while the critical RSB transition point is located at , to be compared with the corresponding Shannon bound . For the binary symmetric channel we show that the low temperature reentrant behavior of the dynamical transition line, observed within the RS ansatz, changes its location when the RSB ansatz is employed; the dynamical transition point occurs at higher values of the channel noise. Possible practical implications to improve the performance of the state-of-the-art error correcting codes are discussed.
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Affiliation(s)
- Gabriele Migliorini
- The Neural Computing Research Group, School of Engineering and Applied Sciences, Aston University, Birmingham B4 7ET, United Kingdom
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Battaglia D, Braunstein A, Chavas J, Zecchina R. Source coding by efficient selection of ground-state clusters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:015103. [PMID: 16090025 DOI: 10.1103/physreve.72.015103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/24/2004] [Indexed: 05/03/2023]
Abstract
We analyze the geometrical structure of clusters of ground states which appear in many frustrated systems over random graphs. Focusing on the regime of connectivities where the number of clusters is exponential in the size of the problems, we identify an appropriate generalization of the survey propagation equations efficiently exploring the geometry. The possibility of selecting different clusters has also computational consequences. As a proof of concept here we show how a well-known physical system can be used to perform nontrivial data compression, for which we introduce a unique compression scheme. Performances are optimized when the number of well-separated clusters is maximal in the underlying physical model.
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15
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Pica Ciamarra M, Tarzia M, de Candia A, Coniglio A. Monodisperse model suitable to study the glass transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:066111. [PMID: 14754273 DOI: 10.1103/physreve.68.066111] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2003] [Indexed: 05/24/2023]
Abstract
We study the properties of a monodisperse lattice glass model with a simple geometrical interpretation, which reproduces many features of glass forming liquids, such as the cage effect, vanishing diffusivity, and the presence of two time scales in relaxation functions. The model has a crystalline ground state at high density, but has no tendency to crystallize when quenched, even at extremely low cooling rates, which makes it suitable for the study of the glass transition. We study the model in mean field on random regular graphs, finding a scenario analogous to p-spin models.
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Affiliation(s)
- M Pica Ciamarra
- Dipartimento di Scienze Fisiche, Università di Napoli Federico II, Via Cintia, I-80126 Naples, Italy
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16
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Cocco S, Dubois O, Mandler J, Monasson R. Rigorous decimation-based construction of ground pure states for spin-glass models on random lattices. PHYSICAL REVIEW LETTERS 2003; 90:047205. [PMID: 12570456 DOI: 10.1103/physrevlett.90.047205] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2002] [Indexed: 05/24/2023]
Abstract
A constructive scheme for determining pure states at very low temperature in the 3-spins glass model on a random lattice is provided, in full agreement with Parisi's one step replica symmetry breaking (RSB) scheme. Proof is based on the analysis of a partial decimation procedure and of the statistical properties of its output, i.e., a reduced Hamiltonian acting on a subset of the initial spins. The number of ground states (GS) in each state, the number of states, and the distances between GS are calculated and correspond to RSB predictions.
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Affiliation(s)
- S Cocco
- CNRS-Laboratoire de Dynamique des Fluides Complexes, 3 Rue de l'Université, 67000 Strasbourg, France
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17
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Mézard M, Zecchina R. Random K-satisfiability problem: from an analytic solution to an efficient algorithm. PHYSICAL REVIEW E 2002; 66:056126. [PMID: 12513575 DOI: 10.1103/physreve.66.056126] [Citation(s) in RCA: 300] [Impact Index Per Article: 13.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2002] [Indexed: 11/07/2022]
Abstract
We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the K=3 satisfiability problem, with very good performances.
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Affiliation(s)
- Marc Mézard
- Laboratoire de Physique Théorique et Modèles Statistiques, CNRS and Université Paris Sud, Bâtiment 100, 91405 Orsay Cedex, France
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18
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Franz S, Leone M, Montanari A, Ricci-Tersenghi F. Dynamic phase transition for decoding algorithms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:046120. [PMID: 12443272 DOI: 10.1103/physreve.66.046120] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2002] [Revised: 08/05/2002] [Indexed: 05/24/2023]
Abstract
The state-of-the-art error correcting codes are based on large random constructions (random graphs, random permutations, etc.) and are decoded by linear-time iterative algorithms. Because of these features, they are remarkable examples of diluted mean-field spin glasses, both from the static and dynamic points of view. We analyze the behavior of decoding algorithms by mapping them onto statistical-physics models. This allows us to understand the intrinsic (i.e., algorithm independent) features of this behavior.
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Affiliation(s)
- S Franz
- International Center for Theoretical Physics, P.O. Box 586, I-34100 Trieste, Italy.
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19
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Mézard M, Parisi G, Zecchina R. Analytic and algorithmic solution of random satisfiability problems. Science 2002; 297:812-5. [PMID: 12089451 DOI: 10.1126/science.1073287] [Citation(s) in RCA: 231] [Impact Index Per Article: 10.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio alpha of clauses to variables less than a threshold alphac are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below alphac, where the proliferation of metastable states is responsible for the onset of complexity in search algorithms. We introduce a class of optimization algorithms that can deal with these metastable states; one such algorithm has been tested successfully on the largest existing benchmark of K-satisfiability.
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Affiliation(s)
- M Mézard
- Laboratoire de Physique Théorique et Modèles Statistiques, CNRS and Université Paris Sud, Bât. 100, 91405 Orsay Cedex, France
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20
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Barthel W, Hartmann AK, Leone M, Ricci-Tersenghi F, Weigt M, Zecchina R. Hiding solutions in random satisfiability problems: a statistical mechanics approach. PHYSICAL REVIEW LETTERS 2002; 88:188701. [PMID: 12005728 DOI: 10.1103/physrevlett.88.188701] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2001] [Indexed: 05/23/2023]
Abstract
A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of statistical mechanics results, we propose random generators of hard and satisfiable instances for the 3-satisfiability problem. The design of the hardest problem instances is based on the existence of a first order ferromagnetic phase transition and the glassy nature of excited states. The analytical predictions are corroborated by numerical results obtained from complete as well as stochastic local algorithms.
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Affiliation(s)
- W Barthel
- Institute for Theoretical Physics, University of Göttingen, Bunsenstrasse 9, 37073 Göttingen, Germany
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21
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Berg J, Mehta A. Glassy dynamics in granular compaction: sand on random graphs. PHYSICAL REVIEW E 2002; 65:031305. [PMID: 11909044 DOI: 10.1103/physreve.65.031305] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2001] [Indexed: 11/07/2022]
Abstract
We discuss the use of a ferromagnetic spin model on a random graph to model granular compaction. A multispin interaction is used to capture the competition between local and global satisfaction of constraints characteristic for geometric frustration. We define an athermal dynamics designed to model repeated taps of a given strength. Amplitude cycling and the effect of permanently constraining a subset of the spins at a given amplitude is discussed. Finally we check the validity of Edwards's hypothesis for the athermal tapping dynamics.
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Affiliation(s)
- Johannes Berg
- Abdus Salam International Centre for Theoretical Physics, 34100 Trieste, Italy
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Crisanti A, Leuzzi L, Parisi G. The 3-SAT problem with large number of clauses in the ∞-replica symmetry breaking scheme. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/3/303] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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23
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Berg J, Sellitto M. Metastable configurations of spin models on random graphs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:016115. [PMID: 11800744 DOI: 10.1103/physreve.65.016115] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2001] [Indexed: 05/23/2023]
Abstract
One-flip stable configurations of an Ising model on a random graph with fluctuating connectivity are examined. In order to perform the quenched average of the number of stable configurations we introduce a global order-parameter function with two arguments. The analytical results are compared with numerical simulations.
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Affiliation(s)
- Johannes Berg
- Abdus Salam International Centre for Theoretical Physics, 34100 Trieste, Italy
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