Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

Boettcher, Stefan

doi:10.1103/physrevlett.124.177202

Cited by 12 publications

(19 citation statements)

References 56 publications

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“…Figure 4 shows the results, where the result obtained by the four-GPU implementation of the above-mentioned MA ( 26) is also shown. The horizontal lines show optimal (dashed) and target (dotted) values estimated using a formula based on statistical mechanics (see section S7 for details) (51,52).…”

Section: Performance For Ultralarge-scale Ising Problemsmentioning

confidence: 99%

High-performance combinatorial optimization based on classical mechanics

et al. 2021

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Quickly obtaining optimal solutions of combinatorial optimization problems has tremendous value but is extremely difficult. Thus, various kinds of machines specially designed for combinatorial optimization have recently been proposed and developed. Toward the realization of higher-performance machines, here, we propose an algorithm based on classical mechanics, which is obtained by modifying a previously proposed algorithm called simulated bifurcation. Our proposed algorithm allows us to achieve not only high speed by parallel computing but also high solution accuracy for problems with up to one million binary variables. Benchmarking shows that our machine based on the algorithm achieves high performance compared to recently developed machines, including a quantum annealer using a superconducting circuit, a coherent Ising machine using a laser, and digital processors based on various algorithms. Thus, high-performance combinatorial optimization is realized by massively parallel implementations of the proposed algorithm based on classical mechanics.

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Section: Performance For Ultralarge-scale Ising Problemsmentioning

confidence: 99%

High-performance combinatorial optimization based on classical mechanics

et al. 2021

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“…We have confirmed this conjecture by means of numerical simulations. The connectivity regime c ∝ N b (0 < b < 1) lies between sparse (c = O(1)) and diluted (c = O(N)) networks [84][85][86][87]. Even though this intermediate connectivity range, called the extremely diluted regime, has been known for a long time in the field of neural networks [95,96], it has been studied only in the case of homogeneous networks, for which degree fluctuations are unimportant.…”

Section: Discussionmentioning

confidence: 99%

“…Figure 7 confirms that the mean-field theory presented in this work describes spin models on heterogeneous networks where c scales as c ∝ N b , with 0 < b < 1. This regime of connectivity lies between sparse networks (b = 0) and diluted networks (b = 1) [84][85][86][87].…”

Section: Random Couplingsmentioning

confidence: 99%

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Metz

Peron

2022

J. Phys. Complex.

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Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-ﬁeld theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-ﬁeld equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local ﬁelds. Traditional mean-ﬁeld theories on fully-connected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington-Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree ﬂuctuations to the phase diagram of mean-ﬁeld spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical ﬁndings and provide compelling evidence that the present mean-ﬁeld theory describes an intermediate regime of connectivity, in which the average degree $c$ scales as a power $c \propto N^{b}$ ($b < 1$) of the total number $N \gg 1$ of spins. Our ﬁndings put forward a novel class of spin models that incorporate the effects of degree ﬂuctuations and, at the same time, are amenable to exact analytic solutions.

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“…We have confirmed this conjecture by means of numerical simulations. The connectivity c ∝ N b (0 < b < 1) lies between sparse (c = O(1)) and diluted (c = O(N )) networks [80][81][82][83]. Even though this intermediate connectivity range, called the extreme diluted regime, has been known for a long time in the field of neural networks [91,92], it has been studied only in the case of homogeneous networks, for which degree fluctuations are unimportant.…”

mentioning

confidence: 99%

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Metz,

Peron

2021

Preprint

View full text Add to dashboard Cite

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the highconnectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fullyconnected models, such as the Curie-Weiss, the Kuramoto, and the Sherrington-Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington-Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

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Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

Cited by 12 publications

References 56 publications

High-performance combinatorial optimization based on classical mechanics

High-performance combinatorial optimization based on classical mechanics

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Mean-field theory of vector spin models on networks with arbitrary degree distributions

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