A mathematical framework for shortest path length computation in multi-layer networks with inter-edge weighting and dynamic inter-edge weighting: The case of the Beijing bus network, China

The search for the shortest paths in some real -world networks with high complexity, essential for practical applications such as map navigation, makes it convenient to consider the influences of dynamic multi -layer and multi -layer networks on the various mathematical models of such networks. In this paper, exact algorithms are presented to accelerate and improve the efficiency of path planning in complex network structures and changing scenarios, thus improving service performance. In particular, Layers and Cross -network Distance Matrices ( LCDM ) method for calculating the shortest path length in multi -layer networks is introduced, the accuracy of which is illustrated with examples and simulations. Through the relationship between the number of cross -nodes and the complexity of LCDM algorithm, we propose the layer sub -network detection processes of large-scale single -layer networks. It is similar to community detection, which makes the application of LCDM algorithm more extensive. In addition, the LCDM-based DLCDM (Dynamic Layers and Cross -network Distance Matrices) algorithm for calculating the shortest path length is introduced. A general case shows the accuracy and the change trend of the shortest path over time. And DLCDM algorithm direct application to the Beijing bus network is shown as a 11 -layer network which, by integrating real-time traffic conditions, enables route planning to respond to traffic jams more flexibly and provide the shortest route based on current traffic conditions, providing drivers with a smoother and more reliable driving experience.