Replica symmetry breaking in weak connectivity systems

Dominicis, C. De; Mottishaw, Peter

doi:10.1088/0305-4470/20/18/009

Cited by 57 publications

(27 citation statements)

References 13 publications

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“…The percolation however does not bother the validity of the replica-symmetric result, which is valid even for percolated non-backbone subgraphs as long as c < e. The proposed structure also allows for a very simple interpretation of approximation (41) of the entropy of minimal VCs: An isolated pair contributes an entropy ln 2 as it has two possible minimal VCs, thus explaining the first term in (41). This entropy is decreased by the insertion of supplementary edges.…”

Section: The Structure Of the Non-backbone Subgraphmentioning

confidence: 89%

“…Despite several efforts, the question of how to handle RSB in finite-connectivity systems is still open. Most attempts [41,42,43,44] try to apply the first step of Parisi's RSB scheme [38] which however was originally developped for infinite-connectivity spin glasses. Due to the more complex structure of the order parameter a complete solution is however still missing.…”

Section: The Simplest One-step Replica Symmetry Broken Solutionmentioning

confidence: 99%

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Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

2001

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The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity c ≃ 1.43. The replica-symmetric solution breaks down at c = e ≃ 2.72. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.

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Section: The Structure Of the Non-backbone Subgraphmentioning

confidence: 89%

Section: The Simplest One-step Replica Symmetry Broken Solutionmentioning

confidence: 99%

Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

2001

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“…This is equivalent to having the correct prior within the Bayesian framework [11] and enables one to obtain analytic expressions for some observables as long as some gauge requirements are obeyed [10]. Numerical solutions show the emergence of stable dominant delta peaks, consistent with those of (8) and (9). The question of longitudinal mode stability (corresponding to the replica symmetric solution) was addressed by setting initial conditions for the numerical solutions close to the solutions (8) and (9), showing that they converge back to these solutions which are therefore stable.…”

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confidence: 88%

Typical Performance of Gallager-Type Error-Correcting Codes

2000

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The performance of Gallager's error-correcting code is investigated via methods of statistical physics. In this approach, the transmitted codeword comprises products of the original message bits selected by two randomly-constructed sparse matrices; the number of non-zero row/column elements in these matrices constitutes a family of codes. We show that Shannon's channel capacity is saturated for many of the codes while slightly lower performance is obtained for others which may be of higher practical relevance. Decoding aspects are considered by employing the TAP approach which is identical to the commonly used belief-propagation-based decoding.The ever increasing information transmission in the modern world is based on communicating messages reliably through noisy transmission channels; these can be telephone lines, deep space, magnetic storing media etc. Errorcorrecting codes play an important role in correcting errors incurred during transmission; this is carried out by encoding the message prior to transmission, and decoding the corrupted received codeword for retrieving the original message. In his ground breaking papers, Shannon [1] analyzed the capacity of communication channels, setting an upper bound to the achievable noise-correction capability of codes, given their code (or symbol) rate. The latter represents the ratio between the number of bits in the original message and the transmitted codeword.Shannon's bound is non-constructive and does not provide explicit rules for devising optimal codes. The quest for more efficient codes, in the hope of saturating the bound set by Shannon, has been going on ever since, providing many useful but sub-optimal codes.One family of codes, presented originally by Gallager In this Letter we analyze the typical performance of Gallager-type codes for several parameter choices via methods of statistical mechanics. We then validate the analytical solution by comparing the results to those obtained by the TAP approach to diluted systems and via numerical methods.In a general scenario, a message represented by an N dimensional Boolean/binary vector ξ is encoded to the M dimensional vector J 0 which is then transmitted through a noisy channel with some flipping probability p per bit (other noise types may also be considered but will not be examined here). The received message J is then decoded to retrieve the original message.One can identify several slightly different versions of Gallager-type codes. The one used in this Letter, termed the MN code [3] is based on choosing two randomly-selected sparse matrices A and B of dimensionality M×N and M×M respectively; these are characterized by K and L non-zero unit elements per row and C and L per column respectively. The finite, usually small, numbers K, C and L define a particular code; both matrices are known to both sender and receiver. Encoding is carried out by constructing the modulo 2 inverse of B and the matrix B −1 A (modulo 2); the vector J 0 = B −1 A ξ (modulo 2, ξ in a Boolean representation) constitutes the codeword....

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“…Such models strongly differ from realistic finite-dimensional systems on two points, that is their large connectivity and the absence of any geometrical underlying structure. As was realized some years ago, only the latter aspect is intrinsic to mean-field theory while the irrealistic nature of the connectivity may be cured [3,[5][6][7][8][9][10]. Mean-field spin-glasses with finite connectivity (FC) are of importance for at least two reasons.…”

Section: Introductionmentioning

confidence: 99%

Optimization problems and replica symmetry breaking in finite connectivity spin glasses

Monasson

1998

J. Phys. A: Math. Gen.

152

220

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A formalism capable of handling the first step of hierarchical replica symmetry breaking in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field distributions (or, equivalently magnetisation distributions) inside the cluster of states. The approach is shown to coincide with the previous works in the replica symmetric case and in the two limit cases m = 0, m = 1 where m is Parisi's break-point. As an application to the study of optimization problems, the ground-state properties of the random 3-Satisfiability problem are investigated and we present a first RSB solution improving replica symmetric results.

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Replica symmetry breaking in weak connectivity systems

Cited by 57 publications

References 13 publications

Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

Typical Performance of Gallager-Type Error-Correcting Codes

Optimization problems and replica symmetry breaking in finite connectivity spin glasses

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