Spectral methods for analyzing and visualizing networks : an introduction

Network analysis begins with data that describes the set of relationships among the members of a system. The goal of analysis is to obtain from the low-level relational data a higher-level description of the structure of the system which identifies various kinds of patterns in the set of relationships. These patterns will be based on the way individuals are related to other individuals in the network. Some approaches to network analysis look for clusters of individuals who are tightly connected to one another; some look for sets of individuals who have similar patterns of relations to the rest of the network. Other methods don’t “look for” anything in particular — instead, they construct a continuous multidimensional representation of the network in which the coordinates of the individuals can be further analyzed to obtain a variety of kinds of information about them and their relation to the rest of the network. One approach to this is to choose a set of axes in the multidimensional space occupied by the network and rotate them so that the first axis points in the direction of the greatest variability in the data; the second axis, orthogonal to the first, points in the direction of greatest remaining variability, and so on. This set of axes is a coordinate system that can be used to describe the relative positions of the set of points in the data. Most of the variability in the locations of points will be accounted for by the first few dimensions of this coordinate system. The coordinates of the points along each axis will be an eigenvector, and the length of the projection will be an eigenvalue. The set of all eigenvalues is the spectrum of the network. Spectral methods (eigendecomposition) have been a part of graph theory for over a century. Network researchers have used spectral methods either implicitly or explicitly since the late 1960's, when computers became generally accessible in most universities. The eigenvalues of a network are intimately connected to important topological features such as maximum distance across the network (diameter), presence of cohesive clusters, long paths and bottlenecks, and how random the network is. The associated eigenvectors can be used as a natural coordinate system for graph visualization; they also provide methods for discovering clusters and other local features. When combined with other, easily obtained network statistics (e.g., node degree), they can be used to describe a variety of network properties, such as degree of robustness (i.e., tolerance to removal of selected nodes or links), and other structural properties, and the relationship of these properties to node or link attributes in large, complex, multivariate networks. We introduce three types of spectral analysis for graphs and describe some of their mathematical properties. We discuss the strengths and weaknesses of each type and show how they can be used to understand network structure. These discussions are accompanied by interactive graphical displays of small (n=50) and moderately large (n=5000) networks. Throughout, we give special attention to sparse matrix methods which allow rapid, efficient storage and analysis of large networks. We briefly describe algorithms and analytic strategies that allow spectral analysis and identification of clusters in very large networks (n>1,000,000).