Complex Stochastic Boolean Systems: Comparing Bitstrings with the Same Hamming Weight

A complex stochastic Boolean system (CSBS) is a complex system depending on an arbitrarily large number n of random Boolean variables. CSBSs arise in many different areas of science and engineering. A proper mathematical model for the analysis of such systems is based on the intrinsic order: a partial order relation defined on the set 0,1n of all binary n-tuples of 0s and 1s. The intrinsic order enables one to compare the occurrence probabilities of two given binary n-tuples with no need to compute them, simply looking at the relative positions of their 0s and 1s. Regarding the analysis of CSBSs, the intrinsic order reduces the complexity of the problem from exponential (2n binary n-tuples) to linear (n Boolean variables). In this paper, using the intrinsic ordering, we compare the occurrence probabilities of any two binary n-tuples having the same number of 1-bits (i.e., the same Hamming weight). Our results can be applied to any CSBS with mutually independent Boolean variables.