Algorithm Comparing Binary String Probabilities In Complex Stochastic Boolean Systems Using Intrinsic Order Graph

This paper deals with a special kind of complex systems which depend on an arbitrary (and usually large) numbernof random Boolean variables. The so-called complex stochastic Boolean systems often appear in many different scientific, technical or social areas. Clearly, there are2nbinary states associated to such a complex system. Each one of them is given by a binary stringu = (u1,…,un) ∈ {0, 1}nofnbits, which has a certain occurrence probabilityPr{u}. The behavior of a complex stochastic Boolean system is determined by the current values of its2nbinaryn-tuple probabilitiesPr{u}and by the ordering between pairs of them. Hence, the intrinsic order graph provides a useful representation of these systems by displaying (scaling) the2nbinaryn-tuples which are ordered in decreasing probability of occurrence. The intrinsic order reduces the complexity of the problem from exponential (2nbinaryn-tuples) to linear (nBoolean variables). For any fixed binaryn-tupleu, this paper presents a new, simple algorithm enabling rapid, elegant determination of all the binaryn-tuplesvwith occurrence probabilities less than or equal to (greater than or equal to)Pr{u}. This algorithm is closely related to the lexicographic (truth-table) order in{0, 1}n, and this is illustrated through the connections (paths) in the intrinsic order graph.