Eigenvectors of directed graphs and importance scores: dominance, T-Rank, and sink remedies

Abstract

We study the properties of the principal eigenvector for the adjacency matrix (and related matrices) for a general directed graph. In particular-motivated by the use of the eigenvector for estimating the "importance" of the nodes in the graph-we focus on the distribution of positive weight in this eigenvector, and give a coherent picture which builds upon and unites earlier results. We also propose a simple method-"T-Rank"-for generating importance scores. T-Rank generates authority scores via a one-level, non-normalized matrix, and is thus distinct from known methods such as PageRank (normalized), HITS (two-level), and SALSA (two-level and normalized). We show, using our understanding of the principal eigenvector, that T-Rank has a much less severe "sink problem" than does PageRank. Also, we offer numerical results which quantify the "tightly-knit community" or TKC effect. We find that T-Rank has a stronger TKC effect than PageRank, and we offer a novel interpolation method which allows for continuous tuning of the strength of this TKC effect. Finally, we propose two new "sink remedies", i.e., methods for ensuring that the principal eigenvector is positive everywhere. One of our sink remedies (source pumping) is unique among sink remedies, in that it gives a positive eigenvector without rendering the graph strongly connected. We offer a preliminary evaluation of the effects and possible applications of these new sink remedies.

From Observability to Significance in Distributed Information Systems

“…The paths most frequently trodden, i.e. have the most frequently observed transitions, or most frequently searched for concepts, become 'classical paths' and may be favoured, someone analogous to a PageRank search [51], [52].…”

On the scaling of functional spaces, from smart cities to cloud computing

“…In reality, a city or community might be partitioned into quite independent regions, leading to a modular reducible form [30]. If one imagines the specific network, which delivers output Y , it may be some maximally quadratic polynomial of N I , related to the structure function of the network, but it may also be significantly less than this.…”

“…One reason is that modularity is only a separation of scales, not an elimination of dependency: dependencies form an ecosystem. Nearest neighbours might hold the greatest semantic importance to a given function, but this reductionist viewpoint is not independently sustained without the eigenstability of the entire web [30].…”

Dependency Centrality from Bipartite Social Networks

“…Although some prominent network theorists have recommended against the use of eigenvector-based centralities on directed data (Bonacich 1972, Valente et al, 2008, such measurements can be conducted on any square matrix, and a number of approaches have been developed to examine in-links and out-links (see Bjelland et al, 2010, for a review). Thus, mathematics does not preclude the calculation of various eigenvector-based centralities on the inherently non-symmetric dependency matrices.…”