Diffusion and consensus on weakly connected directed graphs

Abstract

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.

Three conjectures of Ostrander on digraph Laplacian eigenvectors

“…An eigenfunction of a linear operator can be viewed as a fixed point, namely a timeinvariant point, of the operator in a corresponding projective space. Many dynamical processes on a geometric domain, including diffusion processes and consensus processes [18,24,29,40,42,44,57,58], are driven by a Laplacian operator that reflects the local connectivity scenarios of the space. It is natural to expect that the shape of an eigenfunction of a Laplacian operator should somehow follow the shape of the underlying space; That is, you may be able to tell/predict the shape of space from some time-invariant data.…”

Clustering-Based Model Reduction of Laplacian Dynamics With Weakly Connected Topology

“…The studied systems describe a more general class of networks than undirected networks and strongly connected networks. In practice, weakly connected structures are found in various network applications, e.g., vehicle formations, self-synchronizing sensor networks, pagerank algorithms, and social networks (see [2], [3], [16], and [17]). One of the major difficulties comes from the semistability of the network system, in which the network matrix, different from [14], [15], may contain multiple semisimple zero eigenvalues, and the corresponding eigenvectors may have rows with only zeros.…”

A Primer on Laplacian Dynamics in Directed Graphs

“…Many of the results we will discuss had earlier been "folklore" results living largely outside the mathematics community and not always with complete proofs (see [7,17] for some references). In the mathematics community, directed graphs are still much less studied than undirected graphs (especially true for the algebraic aspects).…”

“…The in-degree Laplacian of G is the same as the out-degree Laplacian for G , the graph G with all orientations reversed. In [17], the convention was proposed where the direction of edges corresponds to the flow of information in the underlying problem. While here we are not discussing any particular applications, we can still make use of that convention.…”