An Efficient and Automatic Simplification Method for Arbitrary Complex Networks in Mine Ventilation

The simplification of complex networks is a research field closely related to graph theory in discrete mathematics. The existing methods are typically limited to simplifying the series sub-networks, parallel sub-networks, diagonal sub-networks, and nested simple sub-networks. From the current perspective, there are no available methods that can handle complex sub-networks and nested complex sub-networks. In this paper, we innovatively propose an efficient and automatic equivalence simplification method for arbitrary complex ventilation networks. The method enables, for the first time, the maximum possible equivalence simplification of nested simple sub-networks and nested complex sub-networks. In order to avoid the NP-hard problem caused by the searching of simplifiable sub-networks, it is necessary to analyze the intrinsic topology relationship between simplifiable sub-networks and spanning sub-graphs to optimize the searching process. One of our main contributions is that we present an efficient searching method for arbitrarily nested reducible sub-networks based on the bidirectional traversal process of a directed tree. The method optimizes the searching process for simplifiable node pairs by combining the characteristics of a directed tree with the judgment rules of simplifiable sub-networks. Moreover, by deriving the formula of an equivalent air resistance calculation for complex sub-networks, another one of our main contributions is that we present an equivalent calculation and simplification method for arbitrarily complex sub-networks based on the principle of energy conservation. The basic idea of the method is to calculate the equivalent air resistance using the ventilation network resolution of the constructed virtual sub-networks. We realize the simplification method of arbitrarily complex mine ventilation networks, and we validate the reliability of the simplification method by comparing the air distribution results using the network solution method before and after simplification. It can be determined that, with appropriate modifications to meet specific requirements, the proposed method can also be applicable to equivalent simplification instances of other types of complex networks. Based on the results analysis of several real-world mine ventilation network examples, the effectiveness of the proposed method is further verified, which can satisfactorily meet the requirements for simplifying complex networks.