2002/04/25 by Leo Kadanoff, Leo P. Kadanoff, S. N. Coppersmith +5 · 3 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Cellular Automata and Applications #Cellular Automata and Lattice Gases (nlin.CG) #FOS: Physical sciences #Gene Regulatory Network Analysis #Stochastic processes and statistical mechanics #nlin.AO #nlin.CG
paper · pdf · doi:10.48550/arxiv.nlin/0204062
69 pages, 16 figures, Submitted to Springer Applied Mathematical Sciences Series
openalex publication_date 2002/04/25 · arxiv created 2002/04/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper reviews a class of generic dissipative dynamical systems called N-K models. In these models, the dynamics of N elements, defined as Boolean variables, develop step by step, clocked by a discrete time variable. Each of the N Boolean elements at a given time is given a value which depends upon K elements in the previous time step. We review the work of many authors on the behavior of the models, looking particularly at the structure and lengths of their cycles, the sizes of their basins of attraction, and the flow of information through the systems. In the limit of infinite N, there is a phase transition between a chaotic and an ordered phase, with a critical phase in between. We argue that the behavior of this system depends significantly on the topology of the network connections. If the elements are placed upon a lattice with dimension d, the system shows correlations related to the standard percolation or directed percolation phase transition on such a lattice. On the other hand, a very different behavior is seen in the Kauffman net in which all spins are equally likely to be coupled to a given spin. In this situation, coupling loops are mostly suppressed, and the behavior of the system is much more like that of a mean field theory. We also describe possible applications of the models to, for example, genetic networks, cell differentiation, evolution, democracy in social systems and neural networks.