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Average number of fixed points and attractors in Hopfield neural networks

Department of Physics and Beijing Key Laboratory of Opto-Electronic, Functional Materials and Micro-Nano Devices, Renmin University Beijing, P. R. China

E-mail Address: liujiandu@ruc.edu.cn

The authors are the first authors who contributed equally to this work.

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Bokui Chen

Division of Logistics and Transportation, Graduate School at Shenzhen, Tsinghua University, Shenzhen, P. R. China

Department of Computer Science, School of Computing, National University of Singapore, Singapore

E-mail Address: chenbk@comp.nus.edu.sg

The authors are the first authors who contributed equally to this work.

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Dengcheng Yan

Department of Modern Physics, University of Science and Technology of China Hefei, P. R. China

E-mail Address: yanzhou@mail.ustc.edu.cn

The authors are the first authors who contributed equally to this work.

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, and

Lei Wang

Department of Physics and Beijing Key Laboratory of Opto-Electronic, Functional Materials and Micro-Nano Devices, Renmin University Beijing, P. R. China

E-mail Address: phywanglei@ruc.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0129183118500766Cited by:1 (Source: Crossref)

Abstract

Calculating the exact number of fixed points and attractors of an arbitrary Hopfield neural network is a non-deterministic polynomial (NP)-hard problem. In this paper, we first calculate the average number of fixed points in such networks versus their size and threshold of neurons, in terms of a statistical method, which has been applied to the calculation of the average number of metastable states in spin glass systems. Then the same method is expanded to study the average number of attractors in such networks. The results of the calculation qualitatively agree well with the numerical calculation. The discrepancies between them are also well explained.

Keywords:

PACS: 07.05.Mh, 07.05.Tp

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