This paper proposes and investigates a Boolean gossip model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossip process defines absorbing Markov chains. Particularly, it is shown that for the majority of the Boolean interaction rules, except for nine out of the total @math possible nonempty sets of binary Boolean functions, whether the induced chain is absorbing has nothing to do with the topology of the underlying interaction graph, as long as connectivity is assumed. These results illustrate possibilities of relating dynamical properties of Boolean networks to graphical properties of the underlying interactions.
Gene Regulation . The evolution of gene expressions can be naturally described as a dynamical system where the two quantized levels, ON and OFF, are represented by logic states 1 and 0, respectively. Each gene normally would only interact with a small number of neighbouring genes Such number is two or three in Kauffman's original proposal @cite_11 . . Therefore, the proposed Boolean gossip network model at least serves as a good approximation for gene regulator networks, where a pair of genes interact at any given time and the Boolean function rules @math describe random outcomes of the interactions.
@cite_11
1971224531
Abstract Proto-organisms probably were randomly aggregated nets of chemical reactions. The hypothesis that contemporary organisms are also randomly constructed molecular automata is examined by modeling the gene as a binary (on-off) device and studying the behavior of large, randomly constructed nets of these binary “genes”. The results suggest that, if each “gene” is directly affected by two or three other “genes”, then such random nets: behave with great order and stability; undergo behavior cycles whose length predicts cell replication time as a function of the number of genes per cell; possess different modes of behavior whose number per net predicts roughly the number of cell types in an organism as a function of its number of genes; and under the stimulus of noise are capable of differentiating directly from any mode of behavior to at most a few other modes of behavior. Cellular differentation is modeled as a Markov chain among the modes of behavior of a genetic net. The possibility of a general theory of metabolic behavior is suggested.