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Boolean Dynamics with
Random Couplings
Maximino Aldana
Susan Coppersmith
Leo P. Kadanoff
To Larry Sirovich on the occasion of his 70th birthday
ABSTRACT This paper reviews a class of generic dissipative dynamical
systems called NK 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
