15,499 research outputs found
The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance
For two decades, the Colless index has been the most frequently used
statistic for assessing the balance of phylogenetic trees. In this article,
this statistic is studied under the Yule and uniform model of phylogenetic
trees. The main tool of analysis is a coupling argument with another well-known
index called the Sackin statistic. Asymptotics for the mean, variance and
covariance of these two statistics are obtained, as well as their limiting
joint distribution for large phylogenies. Under the Yule model, the limiting
distribution arises as a solution of a functional fixed point equation. Under
the uniform model, the limiting distribution is the Airy distribution. The
cornerstone of this study is the fact that the probabilistic models for
phylogenetic trees are strongly related to the random permutation and the
Catalan models for binary search trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000547 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Experimental analysis of the accessibility of drawings with few segments
The visual complexity of a graph drawing is defined as the number of
geometric objects needed to represent all its edges. In particular, one object
may represent multiple edges, e.g., one needs only one line segment to draw two
collinear incident edges. We study the question if drawings with few segments
have a better aesthetic appeal and help the user to asses the underlying graph.
We design an experiment that investigates two different graph types (trees and
sparse graphs), three different layout algorithms for trees, and two different
layout algorithms for sparse graphs. We asked the users to give an aesthetic
ranking on the layouts and to perform a furthest-pair or shortest-path task on
the drawings.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Faster Parametric Shortest Path and Minimum Balance Algorithms
The parametric shortest path problem is to find the shortest paths in graph
where the edge costs are of the form w_ij+lambda where each w_ij is constant
and lambda is a parameter that varies. The problem is to find shortest path
trees for every possible value of lambda.
The minimum-balance problem is to find a ``weighting'' of the vertices so
that adjusting the edge costs by the vertex weights yields a graph in which,
for every cut, the minimum weight of any edge crossing the cut in one direction
equals the minimum weight of any edge crossing the cut in the other direction.
The paper presents fast algorithms for both problems. The algorithms run in
O(nm+n^2 log n) time. The paper also describes empirical studies of the
algorithms on random graphs, suggesting that the expected time for finding a
minimum-mean cycle (an important special case of both problems) is O(n log(n) +
m)
On balanced planar graphs, following W. Thurston
Let be an orientation-preserving branched covering map of
degree , and let be an oriented Jordan curve passing through
the critical values of . Then is an oriented graph
on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced
balanced planar graphs and showed that they combinatorially characterize all
such , where has distinct critical values. We give a
detailed account of this discussion, along with some examples and an appendix
about Hurwitz numbers.Comment: 17 page
Directed nonabelian sandpile models on trees
We define two general classes of nonabelian sandpile models on directed trees
(or arborescences) as models of nonequilibrium statistical phenomena. These
models have the property that sand grains can enter only through specified
reservoirs, unlike the well-known abelian sandpile model.
In the Trickle-down sandpile model, sand grains are allowed to move one at a
time. For this model, we show that the stationary distribution is of product
form. In the Landslide sandpile model, all the grains at a vertex topple at
once, and here we prove formulas for all eigenvalues, their multiplicities, and
the rate of convergence to stationarity. The proofs use wreath products and the
representation theory of monoids.Comment: 43 pages, 5 figures; introduction improve
Statistics of planar graphs viewed from a vertex: A study via labeled trees
We study the statistics of edges and vertices in the vicinity of a reference
vertex (origin) within random planar quadrangulations and Eulerian
triangulations. Exact generating functions are obtained for theses graphs with
fixed numbers of edges and vertices at given geodesic distances from the
origin. Our analysis relies on bijections with labeled trees, in which the
labels encode the information on the geodesic distance from the origin. In the
case of infinitely large graphs, we give in particular explicit formulas for
the probabilities that the origin have given numbers of neighboring edges
and/or vertices, as well as explicit values for the corresponding moments.Comment: 36 pages, 15 figures, tex, harvmac, eps
- …