Abstract
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 models. It is the latter set of applications that necessitates the introduction of vertex weights. Finally, we discuss some open problems and research directions.
Highlights
Introduction and BackgroundThe well-known Erdos–Rényi random graph model [7] has an ordered version introduced in [5] by Barak and Erdos
Takis Konstantopoulos: Research supported by Swedish Research Council Grant 2013-4688
What we do is this: We prove the strong law of large numbers (SLLN) assuming that the vertex weight v is a.s. positive with finite expectation, the edge weight u has positive and finite expectation and that its positive part has finite variance
Summary
The well-known Erdos–Rényi random graph model [7] has an ordered version introduced in [5] by Barak and Erdos. N} and the question of interest is the number of linear extensions of the random partial order Another application of a continuous-vertex extension of SOG(Z, p) appears in the physics literature: Itoh and Krapivsky [16] introduce a version, called “continuum cascade model” of the stochastic ordered graph with set of vertices in R+ and study asymptotics for the length of longest paths between 0 and t > 0, deriving recursive integral equations for its distribution. Studying longest paths is the subject in last passage percolation problems in probability theory In this area, one is given a random directed graph (think of Z+ × Z+ with edges (x, y) where x = (x1, x2) is below y = (y1, y2) component-wise) and random weights on the vertices. We devote the last section to discussing a number of open and exciting new problems that we believe are of interest in several areas of applications of engineering, biology, computer science, stochastic networks, and statistical physics
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