Shared node and its improvement to the theory analysis and solving algorithm for the loop cutset

Bayesian Network is one of the famous network models, and the loop cutset is one of the crucial structures for Bayesian Inference. In the Bayesian Network and its inference, how to measure the relationship between nodes is very important, because the relationship between different nodes has significant influence on the node-probability of the loop cutset. To analyse the relationship between two nodes in a graph, we define the shared node, prove the upper and lower bounds of the shared nodes number, and affirm that the shared node influences the node-probability of the loop cutset according to the theorems and experiments. These results can explain the problems that we found in studying on the statistical node-probability belonging to the loop cutset. The shared nodes are performed not only to improve the theoretical analysis on the loop cutset, but also to the loop cutset solving algorithms, especially the heuristic algorithms, in which the heuristic strategy can be optimized by a shared node. Our results provide a new tool to gauge the relationship between different nodes, a new perspective to estimate the loop cutset, and it is helpful to the loop cutset algorithm and network analysis.