Mapping in Random Structures

A mapping in random structures is defined on the vertices of a generalized hypercube $Q{n \over \alpha}$. A random structure consists of (i) a random contact graph and (ii) a family of relations imposed on adjacent vertices of the random contact graph. The vertex set of a random contact graph is the set of all coordinates of a vertex $V \in Q{n \over \alpha}, {P_1,...,P_n}$. The edges of the random contact graph are the union of the edge sets of two random graphs. The first is a random 1-regular graph on $2m$ vertices ${P_{i_1},...,P_{i_2}m}$ and the second is a random graph $G_p$ with $p = {c_2 \over n}$ on ${P_1,...,P_n}$. The structure of the random contact graphs is investigated and it is shown that for certain values of $m$ and $c_2$ the mapping in random structures allows systematic search in the set of random structures. Finally, the results are applied to evolutionary optimization of biopolymers.