Feed-forward chains of recurrent attractor neural networks with finite dilution near saturation

Metz, Fernando L.; Theumann, W. K.

doi:10.1016/j.physa.2005.11.049

Cited by 2 publications

(3 citation statements)

References 29 publications

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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%

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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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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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show abstract

“…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

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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 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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Feed-forward chains of recurrent attractor neural networks with finite dilution near saturation

Cited by 2 publications

References 29 publications

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

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

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