Dynamical Replica Analysis of Disordered Ising Spin Systems on Finitely Connected Random Graphs

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

doi:10.1103/physrevlett.95.117204

Cited by 29 publications

(61 citation statements)

References 29 publications

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“…According to (B1) the joint probability P (t) (s t i , s t ∂i ) between a given spin and its neighbours at time t can be computed iteratively starting from the initial conditions for P (t) at time zero. Above W i (s t i |s t−1 ∂i ) is the same transition rate for spin i which appears in the main text, whereas T j→(ij) (s t j |s t−1 j , s t−1 i ) represents the two-times message that comes from the 1-step Markov ansatz taken in [6] and which can be computed also iteratively according to equation (13) in [6]. A partial marginalization of (B1) allows then to compute same-time correlations as c t ij = s t i , s t j with j in the neighbourhood of i (j ∈ ∂i), which are though not investigated in this work where we rather focus on correlations at different time.…”

Section: Discussionmentioning

confidence: 99%

“…The results obtained using this second proposal are expected to improve those from the star approximation, the reason being that the factorization in the star anzats is refined by conditioning on the state of the common neighbor, compare (10) and (13 i . This is because one of our working assumptions is that the state of a spin at time t does not depend on the same spin at time t − 1.…”

Section: A Variational Formulation Of Dynamics In Discrete Timementioning

confidence: 99%

“…Many concepts have been introduced in a natural analogy with the equilibrium theory, e.g. dynamical replica analysis [13,14], cavity method [15], dynamic message-passing algorithm [6,7,16,17], large deviation [18,19], TAP approaches [20] and extended Plefka expansion for continuous variables [21]. Despite all these advances, the issue is far from being settled and there is an active community searching for approximate methods that accurately reproduce numerical results from stochastic simulations [22].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A simple analytical description of the non-stationary dynamics in Ising spin systems

Vázquez¹,

Ferraro²,

Ricci-Tersenghi³

2017

J. Stat. Mech.

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The analytical description of the dynamics in models with discrete variables (e.g. Ising spins) is a notoriously difficult problem, that can be tackled only under some approximation. propose to use the same approximation based on the cluster variational method also for the non-stationary regime, which has not been considered up to now within this framework. We check the validity of this approximation in describing the non-stationary dynamical regime of several Ising models defined on Erdos-Rényi random graphs: we study ferromagnetic models with symmetric and partially asymmetric couplings, models with random fields and also spin glass models. A comparison with the actual Glauber dynamics, solved numerically, shows that one of the two studied approximations (the so-called 'diamond' approximation) provides very accurate results in all the systems studied. Only for the spin glass models we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states. Given the simplicity of the equations to be solved, we believe the diamond approximation should be considered as the 'minimal standard' in the description of the non-stationary regime of Ising-like models: any new method pretending to provide a better approximate description to the dynamics of Ising-like models should perform at least as good as the diamond approximation.2

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Section: Discussionmentioning

confidence: 99%

Section: A Variational Formulation Of Dynamics In Discrete Timementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A simple analytical description of the non-stationary dynamics in Ising spin systems

Vázquez¹,

Ferraro²,

Ricci-Tersenghi³

2017

J. Stat. Mech.

View full text Add to dashboard Cite

show abstract

“…Despite some efforts in the last years [25,26,27,28,29], the dynamics of diluted models remains an unsolved problem, and we have to restrict our analysis to approximate methods. Following the ideas of [27], and restricting our attention to the simplest level of description, we can close the equations for e c (t) and e nc (t) by assuming that at each step all configurations of given energy values are equally probable [31].…”

Section: A the Energy Of Canalizing And Non-canalizing Functionsmentioning

confidence: 99%

Propagation of external regulation and asynchronous dynamics in random Boolean networks

Mahmoudi

Pagnani

Weigt

et al. 2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Boolean Networks and their dynamics are of great interest as abstract modeling schemes in various disciplines, ranging from biology to computer science. Whereas parallel update schemes have been studied extensively in past years, the level of understanding of asynchronous updates schemes is still very poor. In this paper we study the propagation of external information given by regulatory input variables into a random Boolean network. We compute both analytically and numerically the time evolution and the asymptotic behavior of this propagation of external regulation (PER). In particular, this allows us to identify variables which are completely determined by this external information. All those variables in the network which are not directly fixed by PER form a core which contains in particular all non-trivial feedback loops. We design a message-passing approach allowing to characterize the statistical properties of these cores in dependence of the Boolean network and the external condition. At the end we establish a link between PER dynamics and the full random asynchronous dynamics of a Boolean network. PACS numbers:The main motivation for this work is to study the propagation of external information given by regulating but non-regulated variables into random Boolean networks. This process, called propagation of external regulation, eventually stops due to one of two reasons: Either the external information has propagated throughout the full network, or a core of variables cannot be fixed by PER. The statistical properties of these cores are determined by the ratio α between the numbers of Boolean constraints and of variables, and by the composition of the constraints. In this work we will embed the propagation dynamics of the external condition into the asynchronous update dynamics by introducing ternary variables of values {0, 1, ⋆}. The supplementary joker value ⋆ indicates that a variable has not been fixed to a Boolean value in the PER dynamics. The introduction of this new value ⋆ allow us to analyze the PER dynamics in terms of a new constraint satisfaction problem with the same topology of the original one, but where Boolean constraints are extended in a natural way in order to include this ternary representation. This observation allows us to apply recently developed tools from the statistical-mechanics approach to combinatorial optimization problems.

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“…Our investigation will exploit the fact that that the understanding of dilute random systems -introduced many years ago by Viana and Bray [40] -has witnessed significant progress in the last few years, regarding their equilibrium properties [41,42,43], as well as their dynamic behaviour [44,45,46,47,48]. This progress was also the basis for the previously discussed lattice-glass models, which, however, due to their discrete nature do not allow for the description of low-temperature excitations.…”

mentioning

confidence: 99%

Finitely coordinated models for low-temperature phases of amorphous systems

Kühn¹,

Mourik²,

Weigt³

et al. 2007

J. Phys. A: Math. Theor.

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We introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite-coordination effects on the low-temperature behavior of finite dimensional systems. Our models use a description in terms of small deviations of particle coordinates from a set of reference positions, particularly appropriate for the description of low-temperature phenomena. A Born-von-Karman type expansion with random coefficients is used to model effects of frozen heterogeneities. The key quantity appearing in the theoretical description is a full distribution of effective single-site potentials which needs to be determined self-consistently. If microscopic interactions are harmonic, the effective single-site potentials turn out to be harmonic as well, and the distribution of these single-site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels. For structural glasses characterized by frustration and anharmonicities in the microscopic interactions, the distribution of single-site potentials involves anharmonicities of all orders, and both single-well and double well potentials are observed, the latter with a broad spectrum of barrier heights. The appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single-site potentials, as previously observed for fully connected systems. Double-well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well known universal glassy low temperature anomalies when quantum effects are taken into account.

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Dynamical Replica Analysis of Disordered Ising Spin Systems on Finitely Connected Random Graphs

Cited by 29 publications

References 29 publications

A simple analytical description of the non-stationary dynamics in Ising spin systems

A simple analytical description of the non-stationary dynamics in Ising spin systems

Propagation of external regulation and asynchronous dynamics in random Boolean networks

Finitely coordinated models for low-temperature phases of amorphous systems

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