Let G be a weakly connected directed graph with asymmetric graph Laplacian L. Consensus and diffusion are dual dynamical processes defined on G byẋ = −Lx for consensus andṗ = −pL for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors {γi} k i=1 of the left null-space of L and a basis of column vectors {γi} k i=1 of the right null-space of 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 (see below) feature usually included in the algorithm is not strictly necessary.Together with [13], 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.