Abstract
Using a probabilistic approach, the parallel dynamics of theQ-state Potts andQ-Ising neural networks are studied at zero and at nonzero temperatures. Evolution equations are derived for the first time step and arbitraryQ. These formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks. An explicit analysis, including dynamical capacity-temperature diagrams and the temperature dependence of the overlap, is carried out forQ=3. Both types of models are compared.
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References
C. Meunier, D. Hansel, and A. Verga,J. Stat. Phys. 55:859 (1989).
J. S. Yedidia,J. Phys. A 22:2265 (1989).
J. Stark and P. Bressloff,J. Phys. A 23:1633 (1990).
H. Rieger,J. Phys. A 23:L1273 (1990).
H. Rieger, inStatistical Mechanics of Neural Networks, L. Garrido, ed. (Springer, Berlin, 1990), p. 33.
D. Bollé, P. Dupont, and J. van Mourik,Europhys. Lett. 15:893 (1991);Physica A 185:357 (1992).
S. Mertens, H. M. Köhler, and S. Bös,J. Phys. A 24:4941 (1991).
G. A. Kohring,J.Stat. Phys. 62:563 (1991).
D. Bollé, P. Dupont, and B. Vinck,J.Phys. A 25:2859 (1992).
M. Bouten and A. Engel, Basin of attraction in networks of multi-state neurons (1992).
A. J. Noest,Phys. Rev. A 38:2196 (1988).
J. Cook,J. Phys. A 22:2057 (1989).
F. Gerl, K. Bauer, and U. Krey, Learning wthQ-state clock neurons: Optimal storage capacity and Ada Tron-algorithm (1992).
A. E. Patrick, P. Picco, J. Ruiz, and V. A. Zagrebnov,Phys. A 24:L637 (1991).
I. Kanter,Phys. Rev. A 37:2739 (1988).
D. Bollé and F. Mallezie,J. Phys. A 22:4409 (1989).
D. Bollé, P. Dupont, and F. Mallezie, inNeural Networks and Spin Glasses, W. K. Theumann and R. Köberle, eds. (World Scientific, Singapore, 1989), p. 151.
D. Bollé and P. Dupont, inStatistical Mechanics of Neural Networks, L. Garrido, ed. (Springer, Berlin, 1990), p. 365.
D. Bollé, P. Dupont, and J. van Mourik,J. Phys. A 24:1065 (1991).
J. P. Nadal and A. Rau,J. Phys. I (Paris)1:1109 (1991).
T. Watkin, A. Rau, D. Bollé, and J. van Mourik,J. Phys. I (Paris)2:167 (1992).
G. M. Shim, D. Kim, and Y. M. Choi,Phys. Rev. A 45:1238 (1992).
D. Bollé, P. Dupont, and J. Huyghebaert,Phys. Rev. A 46:4194 (1992).
P. A. Ferrari, S. Martinez, and P. Picco,J. Stat. Phys. 66:1643 (1992).
V. Gayrard,J. Stat. Phys. 68:977 (1992).
H. Vogt and A. Zippelius,J. Phys. A 25:2209 (1992).
A. E. Patrick and V. A. Zagrebnov,J. Stat. Phys. 63:59 (1991).
V. A. Zagrebnov and A. S. Chvyrov,Sov. Phys.-JETP 68:153 (1989).
E. Barkai, I. Kanter, and H. Sompolinsky,Phys. Rev. A 41:590 (1990).
B. Derrida, E. Gardner, and A. Zippelius,Europhys. Lett. 4:167 (1987).
R. Kree and A. Zippelius, inModels of Neural Networks, E. Domany, J. L. van Hemmen, and K. Schulten, eds. (Springer-Verlag, Berlin, 1991), p. 193.
A. N. Shiryayev,Probability (Springer-Verlag, New York, 1984).
B. Bollobas,Graph Theory: An Introductory Course (Springer-Verlag, New York, 1979).
D. J. Amit, H. Gutfreund, and H. Sompolinsky,Phys. Rev. Lett. 55:1530 (1985);Ann. Phys. (N.Y.)173:30 (1987).
D. Bollé, P. Dupont, and J. Huyghebaert,Physica A 185:363 (1992).
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On leave of absence from the Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia.
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Bollé, D., Vinck, B. & Zagrebnov, V.A. On the parallel dynamics of theQ-state Potts andQ-Ising neural networks. J Stat Phys 70, 1099–1119 (1993). https://doi.org/10.1007/BF01049424
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DOI: https://doi.org/10.1007/BF01049424