Comparison of Spectral Clustering Methods for Graph Models of Pipeline Systems

Abstract

Investigations related to splitting the initial graph into a given number of connected non-intersecting components have found wide practical application. Graph clustering, for example, is used in computer networks, transport, pattern recognition, and in many other areas. Decomposition methods of graph structures make a significant contribution to the performance of search algorithms. It is especially important in conditions of limitations on computing and time resources. And here we should pay special attention to the class of spectral clustering methods that combine elements of graph theory and linear algebra. In this article, we consider the main provisions of the theory of spectral clustering, such as methods of representing a graph in the form of a matrix, their normalization, options for using eigenvectors. The main approaches to normalized spectral clustering of graphs are described: the Shi-Malik (SM) and Ng-Jordan-Weiss (NJW) methods. Decomposition of any graph as a structure with its inherent topology meets the criteria of optimality in connectivity and balance of subgraphs with a small number of clusters. As the number of subdomains increases above a certain value, the probability of incoherent subgraphs appearing in the decomposition structure increases. To solve this problem, we propose an algorithm for the priority distribution of nodes based on the iterative transfer of nodes of isolated regions to the most priority subgraphs-neighbors. The existing methods of spectral decomposition solve different problems from different areas with different success, respectively, well-established methods in solving one problem may be of little use to others. This paper compares the methods of SM and NJW spectral clustering on two graph models of hydraulic networks, for which the criteria for assessing the quality of decomposition of graphs are determined. It is experimentally determined that for both networks the Shi and Malik method is significantly superior to the Ng, Jordan and Weiss method. That makes it more preferable for decomposition of the graph model into connected subdomains.