  Given two subsets A and B of nodes in a directed graph, the conduciveness of
the graph from A to B is the ratio representing how many of the edges outgoing
from nodes in A are incoming to nodes in B. When the graph's nodes stand for
the possible solutions to certain problems of combinatorial optimization,
choosing its edges appropriately has been shown to lead to conduciveness
properties that provide useful insight into the performance of algorithms to
solve those problems. Here we study the conduciveness of CA-rule graphs, that
is, graphs whose node set is the set of all CA rules given a cell's number of
possible states and neighborhood size. We consider several different edge sets
interconnecting these nodes, both deterministic and random ones, and derive
analytical expressions for the resulting graph's conduciveness toward rules
having a fixed number of non-quiescent entries. We demonstrate that one of the
random edge sets, characterized by allowing nodes to be sparsely interconnected
across any Hamming distance between the corresponding rules, has the potential
of providing reasonable conduciveness toward the desired rules. We conjecture
that this may lie at the bottom of the best strategies known to date for
discovering complex rules to solve specific problems, all of an evolutionary
nature.
