Comments on lumping the Google matrix

TL;DR

This paper discusses a method to simplify PageRank calculations by lumping dangling nodes into a single node, offering computational benefits and new theoretical insights through alternative proofs and matrix decomposition perspectives.

Contribution

It provides alternative proofs for existing results on the Google matrix using lumping methods and introduces a new proof approach from matrix decomposition.

Findings

Lumping dangling nodes reduces computational cost.

Alternative proofs validate existing theoretical results.

New matrix decomposition proof offers fresh insights.

Abstract

On the case that the number of dangling nodes is large, PageRank computation can be proceeded with a much smaller matrix through lumping all dangling nodes of a web graph into a single node. Thus, it saves many computational cost and operations. There are also some theoretical contributions on Jordan canonical form of the Google matrix. Motivated by these theoretical contributions, in this note, we provide alternative proofs for some results of Google matrix through the lumping method due to Ipsen and Selee. Specifically we find that the result is also suitable for some subsequent work based on lumping dangling nodes into a node. Besides, an entirely new proof from the matrix decomposition viewpoint is also proposed.