337,014 research outputs found
Average mixing matrix of trees
We investigate the rank of the average mixing matrix of trees, with all
eigenvalues distinct. The rank of the average mixing matrix of a tree on
vertices with distinct eigenvalues is upper-bounded by .
Computations on trees up to vertices suggest that the rank attains this
upper bound most of the times. We give an infinite family of trees whose
average mixing matrices have ranks which are bounded away from this upper
bound. We also give a lower bound on the rank of the average mixing matrix of a
tree.Comment: 18 pages, 2 figures, 3 table
Ubiquity of synonymity: almost all large binary trees are not uniquely identified by their spectra or their immanantal polynomials
There are several common ways to encode a tree as a matrix, such as the
adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of
the natural random walk), and the matrix of pairwise distances between leaves.
Such representations involve a specific labeling of the vertices or at least
the leaves, and so it is natural to attempt to identify trees by some feature
of the associated matrices that is invariant under relabeling. An obvious
candidate is the spectrum of eigenvalues (or, equivalently, the characteristic
polynomial). We show for any of these choices of matrix that the fraction of
binary trees with a unique spectrum goes to zero as the number of leaves goes
to infinity. We investigate the rate of convergence of the above fraction to
zero using numerical methods. For the adjacency and Laplacian matrices, we show
that that the {\em a priori} more informative immanantal polynomials have no
greater power to distinguish between trees
Limit laws of estimators for critical multi-type Galton-Watson processes
We consider the asymptotics of various estimators based on a large sample of
branching trees from a critical multi-type Galton-Watson process, as the sample
size increases to infinity. The asymptotics of additive functions of trees,
such as sizes of trees and frequencies of types within trees, a higher-order
asymptotic of the ``relative frequency'' estimator of the left eigenvector of
the mean matrix, a higher-order joint asymptotic of the maximum likelihood
estimators of the offspring probabilities and the consistency of an estimator
of the right eigenvector of the mean matrix, are established.Comment: Published at http://dx.doi.org/10.1214/105051604000000521 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Correlation, hierarchies, and networks in financial markets
We discuss some methods to quantitatively investigate the properties of
correlation matrices. Correlation matrices play an important role in portfolio
optimization and in several other quantitative descriptions of asset price
dynamics in financial markets. Specifically, we discuss how to define and
obtain hierarchical trees, correlation based trees and networks from a
correlation matrix. The hierarchical clustering and other procedures performed
on the correlation matrix to detect statistically reliable aspects of the
correlation matrix are seen as filtering procedures of the correlation matrix.
We also discuss a method to associate a hierarchically nested factor model to a
hierarchical tree obtained from a correlation matrix. The information retained
in filtering procedures and its stability with respect to statistical
fluctuations is quantified by using the Kullback-Leibler distance.Comment: 37 pages, 9 figures, 3 table
Transfer matrix for spanning trees, webs and colored forests
We use the transfer matrix formalism for dimers proposed by Lieb, and
generalize it to address the corresponding problem for arrow configurations (or
trees) associated to dimer configurations through Temperley's correspondence.
On a cylinder, the arrow configurations can be partitioned into sectors
according to the number of non-contractible loops they contain. We show how
Lieb's transfer matrix can be adapted in order to disentangle the various
sectors and to compute the corresponding partition functions. In order to
address the issue of Jordan cells, we introduce a new, extended transfer
matrix, which not only keeps track of the positions of the dimers, but also
propagates colors along the branches of the associated trees. We argue that
this new matrix contains Jordan cells.Comment: 29 pages, 7 figure
A New Matrix-Tree Theorem
The classical Matrix-Tree Theorem allows one to list the spanning trees of a
graph by monomials in the expansion of the determinant of a certain matrix. We
prove that in the case of three-graphs (that is, hypergraphs whose edges have
exactly three vertices) the spanning trees are generated by the Pfaffian of a
suitably defined matrix. This result can be interpreted topologically as an
expression for the lowest order term of the Alexander-Conway polynomial of an
algebraically split link. We also prove some algebraic properties of our
Pfaffian-tree polynomial.Comment: minor changes, 29 pages, version accepted for publication in Int.
Math. Res. Notice
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