A -Dimensional hypercube

Kauffman ([kauffSO], [kauffOOa]) has introduced a class of parametrizable fitness landscapes called NK-landscapes, that provide a formalism for studying the efficacy of GA evolution as a function of certain statistical properties of the landscape. Given N binary variables Xi = 1, so that x = (xi, X2, , Xjv) represents a vertex of an A -dimensional hypercube, an NK-landscape is defined by a fitness function, JF, of the form... 

In the above, we have given a computational scheme that allows us to define the connections and interactions between components in biochemical networks and to determine the dynamics in the resulting networks. For an arbitrary network, it is not possible to give a precise description of the dynamics without carrying out numerical simulations. However, all the networks obey certain dynamic rules that are set by the stmcture as embodied in the directed A-dimensional hypercube. Moreover, for networks that show certain structural feamres, such as cyclic attractors, it is possible to make precise statements about the dynamics even without further mathematical analysis or simulation. In other cases, analytical techniques are available to give insight into the dynamics observed—for example, in the cases in which it is possible to prove limit cycles... 

To appreciate this latter point, consider the trajectory of a single molecule in space. Let us discretize this trajectory such that we represent the trajectory of the molecule by a succession of regularly spaced points. Thus, these points may be viewed as the p nodes of a cubic lattice. Clearly, in each spatial dimension, 2 out of the totrd number of the p nosingle molecule, it is inunediately clear that we need to replace the original cube by an A -dimensional hypercube such that in eadi dimension the fraction p - 2/p represents the ratio of nodes not on the surface of the hypercube relative to the total number of nodes. To estimate this fraction... 

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