Bridging the Gap Between Probabilistic Logical Networks and Stochastic Logical Networks

The logical network is a framework that describes finite-valued networked systems. It can be applied to many fields, such as Boolean networks, networked evolutionary games, opinion dynamics, and finite automatons, to name but a few. A new kind of logical network, called the stochastic logical network, is proposed in this paper. Compared with probabilistic logical networks, stochastic logical networks can describe more extensive uncertain behavior. First, we prove that a stochastic logical network can be modeled as a non-homogeneous Markov chain under independent conditions and a homogeneous Markov chain under conditionally independent conditions, respectively. Then, a consistency condition is proposed for the non-equivalence between the independent model and the conditionally independent model. This paper proves that a stochastic logical network can be modeled as a homogeneous Markov chain using a power-reducing operator, only under the consistency condition. Finally, connections between probabilistic logical networks and stochastic logical networks are discussed. It is proved that a probabilistic logical network is a special case of stochastic logical networks. Especially, we point out that the reason why the transition matrix of the probabilistic logical network can be obtained using the power-reducing operator is that the probabilistic logical network satisfies the consistency condition.