Abstract
The state-transition matrix description of Kauffman binary networks described in the previous paper is further developed to obtain an analytical expression for the fraction of states involved in limit cycles as a function of the network size and connectivity. The result obtained for totally connected networks agrees with that derived from quite different considerations by other workers. For low connectivity networks the results are in qualitative agreement with the experimental data of Kauffman but there is a quantitative discrepancy which remains to be resolved.
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Literature
Kauffman, S. A. 1969. “Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets.”J. Theor. Biol.,22, 437–467.
Rubin, H. and R. Sitgreave. 1954. “Probability Distributions Related to Random Transformations on a Finite Set.” Tech Report No. 19a. Applied Maths and Stats Lab., Stanford University.
Sherlock, R. A. 1979. “Analysis of the Behaviour of Kauffman Binary Networks: I: State Space Description and the Distribution of Limn Cycle Lengths.”Bull. Math. Biol.,41, 687–705.
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Sherlock, R.A. Analysis of the behaviour of Kauffman binary networks—II. The state cycle fraction for networks of different connectivities. Bltn Mathcal Biology 41, 707–724 (1979). https://doi.org/10.1007/BF02462423
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DOI: https://doi.org/10.1007/BF02462423