Extremal eigenpairs of adjacency matrices wear their sleeves near their hearts: Maximum principles and decay rates for resolving community structure

Abstract

Understanding relational datasets at a high level is a common data mining task and the detection and classification of community structure is one of the foremost algorithmic challenges of data science. A common approach is to model a dataset with a graph and to use the arsenal of graph mining methods to describe the properties of the data and find desired structure. This arsenal includes many numerical linear algebra techniques. A well-known approach is to calculate a few eigenpairs of a matrix associated with the graph and use the information in the eigenvalues and eigenvectors to find diverse properties of the graph. Often these eigenpairs guide graph optimization processes to more efficient nearoptimal solution. For small and quasi-regular graphs, the choice from the buffet of graph-associated matrices is often unimportant as the performance of the technique may not depend much on which graph matrix is employed. However, in large graphs with highly skewed degree distribution, there are several important considerations in this choice. The calculation cost of finding the eigenvectors and the properties that are determined from these eigenvectors both differ dramatically depending which matrix and set of eigenvectors you choose.We present maximum principles and decay rates demonstrating, for scale-free graphs, the extremal eigenvectors of adjacency matrices are fundamentally different than those related to Laplacian matrices. The results suggest that adjacency eigenpairs could be effectively used to detect community structure of a given density involving many mediumto-high-degree vertices, but that their use is likely inappropriate for locating community structure in the low-degree portions of graphs.

Bipartite Communities

“…The HITS algorithm is then spectral clustering using the eigenvectors of A T A. Kleinberg's algorithm is famous for its strength, but it does have a known issue of reporting popular websites instead of websites that are popular in reference to the search query. This is because the large eigenvectors of an adjacency matrix are dominated by vertices of high degree [18], and the normalized Laplacian is known to present results that better match the topology of the graph.…”