Number of Local Attractors of Scale-Free Networks of Hopfield
DOI:
https://doi.org/10.14529/ctcr150401Keywords:
neural networks, Hopfield model, scale-free networks, capacity, attractorAbstract
Estimates of the number of local attractors for the Hopfield model of attractor neural network with continuous time and states where the neuron interaction graph has a scale-free structure are considered. The number of local attractors defines the network capacity, which is an important network characteristic. Numerous works were devoted to the problem of capacity estimations but mainly Boolean networks and the Hopfield models with symmetric interactions were studied.
In the second case the capacity is proportional to the neuron number N. An estimation of the capacity via characteristics of the network interaction graph is found. This estimate implies that the capacity may increase as exp( cNa ), where c, a > 0. Furthermore, a formula, which connects the capacity and the number of strongly connected neurons (hubs) in the network has been found by computer simulations. We show that the logarithm of the capacity is proportional to the hub number and the hub number is proportional to the root of N.
Results can have applications to associative memory neural models and morphogenesis modeling by genetic networks.References
Jeong H., Mason S. P., Barabasi A.L., Otvai Z.N. Lethality and Centrality in Protein Networks. Nature, 2001, vol. 411, pp. 41–42. DOI: 10.1038/35075138
Lesne A. Complex Networks: from Graph Theory to Biology. Letters in Mathematical Physics, 2006, vol. 78, pp. 235–262. DOI: 10.1007/s11005-006-0123-1
Albert R., Barabasi A.L. Statistical Mechanics of Complex Networks. Reviews of Modern Physics, 2002, vol. 74, pp. 47–97. DOI: 10.1103/RevModPhys.74.47
Li X., Cassidy J., Reinke C.A., Fischboeck S., Carthew R.W. A MicroRNA Imparts Robustness against Environmental Fluctuation during Development. Cell, 2009, pp. 273–282.
Bascompte J. Networks in Ecology. Basic and Applied Ecology, 2007, vol. 8, pp. 485–490. DOI: 10.1016/j.baae.2007.06.003
Hirsch M. W., and Smith H. L. Competitive and Cooperative Systems: a Mini-review. Positive Systems, Lecture Notes in Control and Information Sciences, 2003, vol. 294, pp. 183–190. DOI: 10.1007/978-3-540-44928-7_25
Hirsch M. W., Stability and Convergence in Strongly Monotone Dynamical Systems. Journal Fur Die Reine Und Angewandte Mathematik, 1988, vol. 383, pp. 1–58.
Aldana M. Boolean Dynamics of Networks with Scale-free Topology. Physica D: Nonlinear Phenomena, 2003, vol. 185, pp. 45–66. DOI: 10.1016/S0167-2789(03)00174-X
Hopfield J. Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proceedings of the National Academy of Sciences, USA, 1982, vol. 79, pp. 2554–2558.
He L., Hannon G.J. MicroRNAs: Small RNAs with a Big Role in Gene Regulation. Nature Reviews Genetics, 2004, vol. 7, pp. 522–31. DOI: 10.1038/nrg1379
Manu, Surkova S., Spirov A.V., Gursky V.V., Janssens H., Radulescu O., Samsonova M., Sharp D.H., Reinitz J. Canalization of Gene Expression in the Drosophila Blastoderm by Gap Gene Cross Regulation. Plos Biology, 2009, vol. 49, pp. 591–602. DOI: 10.1371/journal.pbio.1000049
Ruelle D. Elements of Differentiable Dynamics and Bifurcation Theory. Academy Press, 1989, p. 32–43.
Vakulenko S., Radulescu O. Flexible and Robust Networks. Fundamental Informatica, 2012, vol. 119, pp. 1–25. DOI: 10.1142/s0219720012410119
Kauffman S.A. Metabolic Stability and Epigenesis in Randomly Constructed Nets. Journal of Theoretical Biology, 1969, vol. 22, pp. 437–467. DOI: 10.1016/0022-5193(69)90015-0
Samuelsson B. and Troein C., Superpolynomial Growth in the Number of Attractors in Kauffman Networks. Physical Review Letters, 2003, vol. 90, no. 3, pp. 098701-1–098701-4.
Vakulenko S.A. A System of Coupled Oscillators Can Have Arbitrary Prescribed Attractors. Journal of Physics A: Mathematical and General, 1994, vol. 7, pp. 2335–2349. DOI: 10.1088/0305-4470/27/7/015
Vakulenko S., Dissipative Systems Generating any Structurally Stable Chaos. Advances in Difference Equations, 2000, vol. 5, pp. 42–80.
