767 research outputs found
Distributed Estimation of Graph Spectrum
In this paper, we develop a two-stage distributed algorithm that enables
nodes in a graph to cooperatively estimate the spectrum of a matrix
associated with the graph, which includes the adjacency and Laplacian matrices
as special cases. In the first stage, the algorithm uses a discrete-time linear
iteration and the Cayley-Hamilton theorem to convert the problem into one of
solving a set of linear equations, where each equation is known to a node. In
the second stage, if the nodes happen to know that is cyclic, the algorithm
uses a Lyapunov approach to asymptotically solve the equations with an
exponential rate of convergence. If they do not know whether is cyclic, the
algorithm uses a random perturbation approach and a structural controllability
result to approximately solve the equations with an error that can be made
small. Finally, we provide simulation results that illustrate the algorithm.Comment: 15 pages, 2 figure
Finite pseudo orbit expansions for spectral quantities of quantum graphs
We investigate spectral quantities of quantum graphs by expanding them as
sums over pseudo orbits, sets of periodic orbits. Only a finite collection of
pseudo orbits which are irreducible and where the total number of bonds is less
than or equal to the number of bonds of the graph appear, analogous to a cut
off at half the Heisenberg time. The calculation simplifies previous approaches
to pseudo orbit expansions on graphs. We formulate coefficients of the
characteristic polynomial and derive a secular equation in terms of the
irreducible pseudo orbits. From the secular equation, whose roots provide the
graph spectrum, the zeta function is derived using the argument principle. The
spectral zeta function enables quantities, such as the spectral determinant and
vacuum energy, to be obtained directly as finite expansions over the set of
short irreducible pseudo orbits.Comment: 23 pages, 4 figures, typos corrected, references added, vacuum energy
calculation expande
Convex Graph Invariant Relaxations For Graph Edit Distance
The edit distance between two graphs is a widely used measure of similarity
that evaluates the smallest number of vertex and edge deletions/insertions
required to transform one graph to another. It is NP-hard to compute in
general, and a large number of heuristics have been proposed for approximating
this quantity. With few exceptions, these methods generally provide upper
bounds on the edit distance between two graphs. In this paper, we propose a new
family of computationally tractable convex relaxations for obtaining lower
bounds on graph edit distance. These relaxations can be tailored to the
structural properties of the particular graphs via convex graph invariants.
Specific examples that we highlight in this paper include constraints on the
graph spectrum as well as (tractable approximations of) the stability number
and the maximum-cut values of graphs. We prove under suitable conditions that
our relaxations are tight (i.e., exactly compute the graph edit distance) when
one of the graphs consists of few eigenvalues. We also validate the utility of
our framework on synthetic problems as well as real applications involving
molecular structure comparison problems in chemistry.Comment: 27 pages, 7 figure
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