16,933 research outputs found
Spectral Theory for Networks with Attractive and Repulsive Interactions
There is a wealth of applied problems that can be posed as a dynamical system
defined on a network with both attractive and repulsive interactions. Some
examples include: understanding synchronization properties of nonlinear
oscillator;, the behavior of groups, or cliques, in social networks; the study
of optimal convergence for consensus algorithm; and many other examples.
Frequently the problems involve computing the index of a matrix, i.e. the
number of positive and negative eigenvalues, and the dimension of the kernel.
In this paper we consider one of the most common examples, where the matrix
takes the form of a signed graph Laplacian. We show that the there are
topological constraints on the index of the Laplacian matrix related to the
dimension of a certain homology group. In certain situations, when the homology
group is trivial, the index of the operator is rigid and is determined only by
the topology of the network and is independent of the strengths of the
interactions. In general these constraints give upper and lower bounds on the
number of positive and negative eigenvalues, with the dimension of the homology
group counting the number of eigenvalue crossings. The homology group also
gives a natural decomposition of the dynamics into "fixed" degrees of freedom,
whose index does not depend on the edge-weights, and an orthogonal set of
"free" degrees of freedom, whose index changes as the edge weights change. We
also present some numerical studies of this problem for large random matrices.Comment: 27 pages; 9 Figure
Integer symmetric matrices having all their eigenvalues in the interval [-2,2]
We completely describe all integer symmetric matrices that have all their
eigenvalues in the interval [-2,2]. Along the way we classify all signed
graphs, and then all charged signed graphs, having all their eigenvalues in
this same interval. We then classify subsets of the above for which the integer
symmetric matrices, signed graphs and charged signed graphs have all their
eigenvalues in the open interval (-2,2).Comment: 33 pages, 18 figure
Flow polytopes of signed graphs and the Kostant partition function
We establish the relationship between volumes of flow polytopes associated to
signed graphs and the Kostant partition function. A special case of this
relationship, namely, when the graphs are signless, has been studied in detail
by Baldoni and Vergne using techniques of residues. In contrast with their
approach, we provide entirely combinatorial proofs inspired by the work of
Postnikov and Stanley on flow polytopes. As a fascinating special family of
flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the
beautiful volume formula for the type version,
where is the th Catalan number, we introduce type and
Chan-Robbins-Yuen polytopes along with intriguing conjectures
pertaining to their properties.Comment: 29 pages, 13 figure
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