Diffusion and consensus on weakly connected directed graphs

TL;DR

This paper provides a comprehensive analysis of diffusion and consensus processes on weakly connected directed graphs, characterizing their asymptotic behavior and offering new insights into the PageRank algorithm without teleportation.

Contribution

It introduces a complete characterization of diffusion and consensus on directed graphs using null-space eigenvectors and presents a dual perspective on PageRank, removing the need for teleportation.

Findings

Complete asymptotic characterization of diffusion and consensus.

Dual treatment of PageRank without teleportation.

Accessible survey of related mathematical ideas.

Abstract

Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian ${\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$ by $\dot x=-{\cal L}x$ for consensus and $\dot p=-p{\cal L}$ for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors $\{\bar \gamma_i\}_{i=1}^k$ of the left null-space of ${\cal L}$ and a basis of column vectors $\{\gamma_i\}_{i=1}^k$ of the right null-space of ${\cal L}$ in terms of the partition of $G$ into strongly connected components. This allows for complete characterization of the asymptotic behavior of both diffusion and consensus --- discrete and continuous --- in terms of these eigenvectors. As an application of these ideas, we present a treatment of the pagerank algorithm that is dual to the usual one. We further show that the teleporting feature usually included in the algorithm is not strictly necessary. This is a complete and self-contained treatment of the asymptotics of consensus and diffusion on digraphs. Many of the ideas presented here can be found scattered in the literature, though mostly outside mainstream mathematics and not always with complete proofs. This paper seeks to remedy this by providing a compact and accessible survey.