11 research outputs found
Random Sequential Generation of Intervals for the Cascade Model of Food Webs
The cascade model generates a food web at random. In it the species are
labeled from 0 to , and arcs are given at random between pairs of the
species. For an arc with endpoints and (), the species is
eaten by the species labeled . The chain length (height), generated at
random, models the length of food chain in ecological data. The aim of this
note is to introduce the random sequential generation of intervals as a Poisson
model which gives naturally an analogous behavior to the cascade model
Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis
We study a class of directed random graphs. In these graphs, the interval
[0,x] is the vertex set, and from each y\in [0,x], directed links are drawn to
points in the interval (y,x] which are chosen uniformly with density one. We
analyze the length of the longest directed path starting from the origin. In
the large x limit, we employ traveling wave techniques to extract the
asymptotic behavior of this quantity. We also study the size of a cascade tree
composed of vertices which can be reached via directed paths starting at the
origin.Comment: 12 pages, 2 figures; figure adde
Convergence to the Tracy-Widom distribution for longest paths in a directed random graph
We consider a directed graph on the 2-dimensional integer lattice, placing a
directed edge from vertex to , whenever ,
, with probability , independently for each such pair of
vertices. Let denote the maximum length of all paths contained in an
rectangle. We show that there is a positive exponent , such
that, if , as , then a properly centered/rescaled
version of converges weakly to the Tracy-Widom distribution. A
generalization to graphs with non-constant probabilities is also discussed.Comment: 20 pages, 2 figure
Limit theorems for a random directed slab graph
We consider a stochastic directed graph on the integers whereby a directed
edge between and a larger integer exists with probability
depending solely on the distance between the two integers. Under broad
conditions, we identify a regenerative structure that enables us to prove limit
theorems for the maximal path length in a long chunk of the graph. The model is
an extension of a special case of graphs studied by Foss and Konstantopoulos,
Markov Process and Related Fields, 9, 413-468. We then consider a similar type
of graph but on the `slab' , where is a finite partially
ordered set. We extend the techniques introduced in the in the first part of
the paper to obtain a central limit theorem for the longest path. When is
linearly ordered, the limiting distribution can be seen to be that of the
largest eigenvalue of a random matrix in the Gaussian unitary
ensemble (GUE).Comment: 26 pages, 3 figure
Long-range last-passage percolation on the line
We consider directed last-passage percolation on the random graph G = (V,E)
where V = Z and each edge (i,j), for i < j, is present in E independently with
some probability 0 < p <= 1. To every present edge (i,j) we attach i.i.d.
random weights v_{i,j} > 0. We are interested in the behaviour of w_{0,n},
which is the maximum weight of all directed paths from 0 to n, as n tends to
infinity. We see two very different types of behaviour, depending on whether
E[v_{i,j}^2] is finite or infinite. In the case where E[v_{i,j}^2] is finite we
show that the process has a certain regenerative structure, and prove a strong
law of large numbers and, under an extra assumption, a functional central limit
theorem. In the situation where E[v_{i,j}^2] is infinite we obtain scaling laws
and asymptotic distributions expressed in terms of a "continuous last-passage
percolation" model on [0,1]; these are related to corresponding results for
two-dimensional last-passage percolation with heavy-tailed weights obtained by
Hambly and Martin.Comment: 38 pages. Accepted by Annals of Applied Probabilit
Limiting Properties of Random Graph Models with Vertex and Edge Weights
This paper provides an overview of results, concerning longest or heaviest
paths, in the area of random directed graphs on the integers along with some
extensions. We study first-order asymptotics of heaviest paths allowing weights
both on edges and vertices and assuming that weights on edges are signed. We
aim at an exposition that summarizes, simplifies, and extends proof ideas. We
also study sparse graph asymptotics, showing convergence of the weighted random
graphs to a certain weighted graph that can be constructed in terms of Poisson
processes. We are motivated by numerous applications, ranging from ecology to
parallel computing model. It is the latter set of applications that
necessitates the introduction of vertex weights. Finally, we discuss some open
problems and research directions
Limit theorems for the maximal path weight in a directed graph on the line with random weights of edges
We consider the infinite directed graph with vertices the set of integers
...,-2,-1,0,1,2,... . Let v be a random variable taking either finite values or
value "minus infinity". Consider random weights v(j,k), indexed by pairs (j,k)
of integers with j<k, and assume that they are i.i.d. copies of v. The set of
edges of the graph is the set (j,k), j<k. A path in the graph from vertex j to
vertex k, j<k, is a finite sequence of edges (j(0), j(1)), (j(1), j(2)), ...,
(j(m-1), j(m)) with j(0)=j and j(m)=j; the weight of this path is taken to be
the sum v(j(0),j(1))+v(j(1),j(2))+...+v(j(m-1),j(m)) of the weights of its
edges. Let w(0,n) be the maximal weight of all paths from 0 to n. We study the
asymptotic behaviour of the sequence w(0,n), n=1, 2, ..., as n tends to
infinity, under the assumptions that P(v>0)>0, the conditional distribution of
v, given v>0, is not degenerate, and that E exp(Cv) is finite, for some C>0. We
derive local limit theorems in the normal and moderate large deviations regimes
in the case where v has an arithmetic distribution. We also derive an
integro-local theorem in the case where v has a non-lattice distribution.Comment: 16 pages, 1 figur
Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation
Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution supported on with essential supremum equal to (a charge of is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by . Even in the simplest case where , corresponding to the longest path in the Barak-Erd\H{o}s random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call "Max Growth System" (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant . Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional
Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph
To each edge (i,j), i<j of the complete directed graph on the integers we
assign unit weight with probability p or weight x with probability 1-p,
independently from edge to edge, and give to each path weight equal to the sum
of its edge weights. If W^x_{0,n} is the maximum weight of all paths from 0 to
n then W^x_{0,n}/n \to C_p(x),as n\to\infty, almost surely, where C_p(x) is
positive and deterministic. We study C_p(x) as a function of x, for fixed 0<p<1
and show that it is a strictly increasing convex function that is not
differentiable if and only if x is a nonpositive rational or a positive integer
except 1 or the reciprocal of it. We allow x to be any real number, even
negative, or, possibly, -\infty. The case x=-\infty corresponds to the
well-studied directed version of the Erd"os-R'enyi random graph (known as
Barak-Erd"os graph) for which C_p(-\infty) = lim_{x\to -\infty} C_p(x) has been
studied as a function of p in a number of papers.Comment: 24 pages, 6 figure