Ratio-based distortion and network distance

Due to the prevalent theoretical status and high-performance efficiency in computer networks, face recognition, and heterogeneous data analysis, network distance has received mass attention and become an emerging technology in recent years. It is noted that the original difference-based network distance is incompetent to describe the network discrepancy by proportionally expanding/contracting the weight function. To mitigate this shortcoming, this paper proposes ratio-based distortion and network distance, and defines a proportional strong/weak isomorphism which is compatible with the new setting. Several results with theoretical underpinning are deduced by means of mathematical analysis approaches. Additionally, this paper conducts a similarity analysis experiment on a randomly generated network structure dataset using the proposed network distance formula. The analysis of the experimental results, including the corresponding hierarchical clustering diagrams and heatmaps, indicates that the proposed network distance formula has practical application value. The code and data of this paper are completely public at https://github.com/AizhEngHN/Ratio-based-distortion-and-network-distance. 2010 Mathematics Subject Classification: 68T09, 54E35.