150 research outputs found
Second-Order Agents on Ring Digraphs
The paper addresses the problem of consensus seeking among second-order
linear agents interconnected in a specific ring topology. Unlike the existing
results in the field dealing with one-directional digraphs arising in various
cyclic pursuit algorithms or two-directional graphs, we focus on the case where
some arcs in a two-directional ring graph are dropped in a regular fashion. The
derived condition for achieving consensus turns out to be independent of the
number of agents in a network.Comment: 6 pages, 10 figure
Diffusion and consensus on weakly connected directed graphs
Let be a weakly connected directed graph with asymmetric graph Laplacian
. Consensus and diffusion are dual dynamical processes defined on
by for consensus and for diffusion. We
consider both these processes as well their discrete time analogues. We define
a basis of row vectors of the left null-space of
and a basis of column vectors of the right
null-space of in terms of the partition of 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.Comment: 19 pages, Survey Article, 1 figur
Isospectral Graph Reductions and Improved Estimates of Matrices' Spectra
Via the process of isospectral graph reduction the adjacency matrix of a
graph can be reduced to a smaller matrix while its spectrum is preserved up to
some known set. It is then possible to estimate the spectrum of the original
matrix by considering Gershgorin-type estimates associated with the reduced
matrix. The main result of this paper is that eigenvalue estimates associated
with Gershgorin, Brauer, Brualdi, and Varga improve as the matrix size is
reduced. Moreover, given that such estimates improve with each successive
reduction, it is also possible to estimate the eigenvalues of a matrix with
increasing accuracy by repeated use of this process.Comment: 32 page
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