213 research outputs found
Entropy of Some Models of Sparse Random Graphs With Vertex-Names
Consider the setting of sparse graphs on N vertices, where the vertices have
distinct "names", which are strings of length O(log N) from a fixed finite
alphabet. For many natural probability models, the entropy grows as cN log N
for some model-dependent rate constant c. The mathematical content of this
paper is the (often easy) calculation of c for a variety of models, in
particular for various standard random graph models adapted to this setting.
Our broader purpose is to publicize this particular setting as a natural
setting for future theoretical study of data compression for graphs, and (more
speculatively) for discussion of unorganized versus organized complexity.Comment: 31 page
The Incipient Giant Component in Bond Percolation on General Finite Weighted Graphs
On a large finite connected graph let edges become "open" at independent
random Exponential times of arbitrary rates . Under minimal assumptions,
the time at which a giant component starts to emerge is weakly concentrated
around its mean
Weak Concentration for First Passage Percolation Times on Graphs and General Increasing Set-valued Processes
A simple lemma bounds for hitting times
in Markov chains with a certain strong monotonicity property. We show how this
lemma may be applied to several increasing set-valued processes. Our main
result concerns a model of first passage percolation on a finite graph, where
the traversal times of edges are independent Exponentials with arbitrary rates.
Consider the percolation time between two arbitrary vertices. We prove that
is small if and only if is
small, where is the maximal edge-traversal time in the percolation path
attaining
How to Combine Fast Heuristic Markov Chain Monte Carlo with Slow Exact Sampling
Use each of n exact samples as the initial state for a MCMC sampler run for m
steps. We give confidence intervals for accuracy of estimators which are always
valid and which, in certain settings, are almost as good as the intervals one
would obtain if the (unknown) mixing time of the chain were known.Comment: 14 page
A survey of max-type recursive distributional equations
In certain problems in a variety of applied probability settings (from
probabilistic analysis of algorithms to statistical physics), the central
requirement is to solve a recursive distributional equation of the form X =^d
g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are
independent copies of the unknown distribution X. We survey this area,
emphasizing examples where the function g(\cdot) is essentially a ``maximum''
or ``minimum'' function. We draw attention to the theoretical question of
endogeny: in the associated recursive tree process X_i, are the X_i measurable
functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A critical branching process model for biodiversity
Motivated as a null model for comparison with data, we study the following
model for a phylogenetic tree on extant species. The origin of the clade is
a random time in the past, whose (improper) distribution is uniform on
. After that origin, the process of extinctions and speciations is
a continuous-time critical branching process of constant rate, conditioned on
having the prescribed number of species at the present time. We study
various mathematical properties of this model as limits: time of
origin and of most recent common ancestor; pattern of divergence times within
lineage trees; time series of numbers of species; number of extinct species in
total, or ancestral to extant species; and "local" structure of the tree
itself. We emphasize several mathematical techniques: associating walks with
trees, a point process representation of lineage trees, and Brownian limits.Comment: 31 pages, 7 figure
Connected Spatial Networks over Random Points and a Route-Length Statistic
We review mathematically tractable models for connected networks on random
points in the plane, emphasizing the class of proximity graphs which deserves
to be better known to applied probabilists and statisticians. We introduce and
motivate a particular statistic measuring shortness of routes in a network.
We illustrate, via Monte Carlo in part, the trade-off between normalized
network length and in a one-parameter family of proximity graphs. How close
this family comes to the optimal trade-off over all possible networks remains
an intriguing open question. The paper is a write-up of a talk developed by the
first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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