Weak convergence of random p-mappings and the exploration process of inhomogeneous continuum random trees

We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2003) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of ``attracting points'' to emerge.Comment: 16 page

Optimal flow through the disordered lattice

Consider routing traffic on the N x N torus, simultaneously between all source-destination pairs, to minimize the cost $\sum_ec(e)f^2(e)$, where f(e) is the volume of flow across edge e and the c(e) form an i.i.d. random environment. We prove existence of a rescaled $N\to \infty$ limit constant for minimum cost, by comparison with an appropriate analogous problem about minimum-cost flows across a M x M subsquare of the lattice.Comment: Published at http://dx.doi.org/10.1214/009117906000000719 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spatial Transportation Networks with Transfer Costs: Asymptotic Optimality of Hub and Spoke Models

Consider networks on $n$ vertices at average density 1 per unit area. We seek a network that minimizes total length subject to some constraint on journey times, averaged over source-destination pairs. Suppose journey times depend on both route-length and number of hops. Then for the constraint corresponding to an average of 3 hops, the length of the optimal network scales as $n^{13/10}$. Alternatively, constraining the average number of hops to be 2 forces the network length to grow slightly faster than order $n^{3/2}$. Finally, if we require the network length to be O(n) then the mean number of hops grows as order $\log \log n$. Each result is an upper bound in the worst case (of vertex positions), and a lower bound under randomness or equidistribution assumptions. The upper bounds arise in simple hub and spoke models, which are therefore optimal in an order of magnitude sense

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 $e$ become "open" at independent random Exponential times of arbitrary rates $w_e$. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around its mean

Self-similar fragmentations derived from the stable tree II: splitting at nodes

We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Levy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin.Comment: 32 page

Weak Concentration for First Passage Percolation Times on Graphs and General Increasing Set-valued Processes

A simple lemma bounds $\mathrm{s.d.}(T)/\mathbb{E} T$ for hitting times $T$ 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 $X$ between two arbitrary vertices. We prove that $\mathrm{s.d.}(X)/\mathbb{E} X$ is small if and only if $\Xi/\mathbb{E} X$ is small, where $\Xi$ is the maximal edge-traversal time in the percolation path attaining $X$

The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study

Consider a network linking the points of a rate-$1$ Poisson point process on the plane. Write \Psi^{\mbox{ave}}(s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most $s$ times the Euclidean distance. We give upper and lower bounds on the function \Psi^{\mbox{ave}}(s), and on the analogous "worst-case" function \Psi^{\mbox{worst}}(s) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent $\alpha$ such that each function has $\Psi(s) \asymp (s-1)^{-\alpha}$ as $s \downarrow 1$.Comment: 33 page