Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

Hatchett, J. P. L.; Wemmenhove, B.; Castillo, Isaac Pérez; Nikoletopoulos, T; Skantzos, N. S.; Coolen, A. C. C.

doi:10.1088/0305-4470/37/24/001

Cited by 50 publications

(102 citation statements)

References 33 publications

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“…However, the structure of (3) is similar to the one that describes evolution of the disordered Ising spin system [14]. This similarity becomes more apparent if one regards the layers in our model as discrete time-steps of parallel dynamics.…”

Section: Discussionmentioning

confidence: 66%

On the robustness of random Boolean formulae

Mozeika

Saad

Raymond

2010

J. Phys.: Conf. Ser.

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Abstract. Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models.

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Section: Discussionmentioning

confidence: 66%

On the robustness of random Boolean formulae

Mozeika

Saad

Raymond

2010

J. Phys.: Conf. Ser.

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“…In particular due to the unexpectedly rich and varied range of multi-disciplinary applications of finite connectivity replica techniques which emerged subsequently in, for example, spin-glass modelling [6][7][8][9], error correcting codes [10][11][12][13], theoretical computer science [14][15][16][17], recurrent neural networks [18][19][20] and 'small-world' networks [21], this field is presently enjoying a renewed interest and popularity. Until very recently, analysis was limited to the equilibrium properties of such models, but now attention has also turned to the dynamics of finitely connected spin systems [22][23][24][25], using combinatorial and generating functional methods. In the domain of physical spin systems, research into finitely connected systems has usually been triggered by the desire to develop solvable spinglass models which are closer to real finite-dimensional systems than the celebrated fully connected spin-glass model of [26].…”

Section: Introductionmentioning

confidence: 99%

Finitely connected vector spin systems with random matrix interactions

Coolen¹,

Skantzos²,

Castillo³

et al. 2005

J. Phys. A: Math. Gen.

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We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Finitely connected spin models, although still of a mean-field nature, can be regarded as a convenient level of description in between fully connected and finite-dimensional ones. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d = 2 (finitely connected XY spins with random chiral interactions) and for d = 3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

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“…The main idea behind the dynamic belief propagation is to write usual cavity equations using the time trajectories of nodes as variables. This idea has been exploited in a number of previous works on the dynamics of disordered systems [18][19][20][21].…”

Section: Dynamic Belief Propagationmentioning

confidence: 99%

“…One category of problems that has recently attracted a lot of attention is the case of out-of-equilibrium dynamic processes on sparse graphs [18][19][20][21]. Methods which are developed in this context can also be used as so- * Electronic address: andrey.lokhov@lptms.u-psud.fr phisticated mean-field-type approximations for problems defined on general graphs.…”

Section: Introductionmentioning

confidence: 99%

Dynamic message-passing equations for models with unidirectional dynamics

2015

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Understanding and quantifying the dynamics of disordered out-of-equilibrium models is an important problem in many branches of science. Using the dynamic cavity method on time trajectories, we construct a general procedure for deriving the dynamic message-passing equations for a large class of models with unidirectional dynamics, which includes the zero-temperature random field Ising model, the susceptible-infected-recovered model, and rumor spreading models. We show that unidirectionality of the dynamics is the key ingredient that makes the problem solvable. These equations are applicable to single instances of the corresponding problems with arbitrary initial conditions, and are asymptotically exact for problems defined on locally tree-like graphs. When applied to real-world networks, they generically provide a good analytic approximation of the real dynamics.

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Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

Cited by 50 publications

References 33 publications

On the robustness of random Boolean formulae

On the robustness of random Boolean formulae

Finitely connected vector spin systems with random matrix interactions

Dynamic message-passing equations for models with unidirectional dynamics

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