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David J. Aldous

Publications and source records attributed to David J. Aldous.

32 records · Page 2Linked to original sources
arXiv2010

More Uses of Exchangeability: Representations of Complex Random Structures

We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum random trees; second-order limits of distances in random graphs; isometry classes of metric spaces with probability measures; limits of dense random graphs; and more sophisticated uses in finitary combinatorics.

math.PR↗
arXiv2009

Which Connected Spatial Networks on Random Points have Linear Route-Lengths?

In a model of a connected network on random points in the plane, one expects that the mean length of the shortest route between vertices at distance $r$ apart should grow only as $O(r)$ as $r \to \infty$, but this is not always easy to verify. We give a general sufficient condition for such linearity, in the setting of a Poisson point process. In a $L \times L$ square, define a subnetwork $\GG_L$ to have the edges which are present regardless of the configuration outside the square; the condition is that the largest component of $\GG_L$ should contain a proportion $1 - o(1)$ of the vertices, as $L \to \infty$. The proof is by comparison with oriented percolation. We show that the general result applies to the relative neighborhood graph, and establishing the linearity property for this network immediately implies it for a large family of proximity graphs.

math.PR↗
arXiv2009

The Shape Theorem for Route-lengths in Connected Spatial Networks on Random Points

For a connected network on Poisson points in the plane, consider the route-length $D(r,θ) $ between a point near the origin and a point near polar coordinates $(r,θ)$, and suppose $E D(r,θ) = O(r)$ as $r \to \infty$. By analogy with the shape theorem for first-passage percolation, for a translation-invariant and ergodic network one expects $r^{-1} D(r, θ)$ to converge as $r \to \infty$ to a constant $ρ(θ)$. It turns out there are some subtleties in making a precise formulation and a proof. We give one formulation and proof via a variant of the subadditive ergodic theorem wherein random variables are sometimes infinite.

math.PR↗
arXiv2008

Optimal spatial transportation networks where link-costs are sublinear in link-capacity

Consider designing a transportation network on $n$ vertices in the plane, with traffic demand uniform over all source-destination pairs. Suppose the cost of a link of length $\ell$ and capacity $c$ scales as $\ell c^β$ for fixed $0<β<1$. Under appropriate standardization, the cost of the minimum cost Gilbert network grows essentially as $n^{α(β)}$, where $α(β) = 1 - \fracβ{2}$ on $0 < β\leq {1/2}$ and $α(β) = {1/2} + \fracβ{2}$ on ${1/2} \leq β< 1$. This quantity is an upper bound in the worst case (of vertex positions), and a lower bound under mild regularity assumptions. Essentially the same bounds hold if we constrain the network to be efficient in the sense that average route-length is only $1 + o(1)$ times average straight line length. The transition at $β= {1/2}$ corresponds to the dominant cost contribution changing from short links to long links. The upper bounds arise in the following type of hierarchical networks, which are therefore optimal in an order of magnitude sense. On the large scale, use a sparse Poisson line process to provide long-range links. On the medium scale, use hierachical routing on the square lattice. On the small scale, link vertices directly to medium-grid points. We discuss one of many possible variant models, in which links also have a designed maximum speed $s$ and the cost becomes $\ell c^βs^γ$.

cond-mat.dis-nn↗
arXiv2007

Short-length routes in low-cost networks via Poisson line patterns

In designing a network to link n cities in a square of area n, one might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all cities. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two differences? For typical configurations the shortest network length is order n and the average straight-line distance is order n^1/2, so it seems implausible that one can construct a network in which the first difference is o(n) and the second difference is o(n^1/2). But in fact one can do better: for an arbitrary configuration one can construct a network where the first difference is o(n) and the second difference is almost as small as O(log n). The construction is conceptually simple: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. The key ingredient is a new result about the Poisson line process. Consider two points at distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length 2r + C log r. Turning to lower bounds we show that, under a weak equidistribution assumption, if the first difference is O(n) then the second difference cannot be o(sqrt(log n)).

math.PR↗
arXiv2007

Dynamic Programming Optimization over Random Data: the Scaling Exponent for Near-optimal Solutions

A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over sets A in {1,2,...,n}, the objective function |A| - \sum_i ξ_i 1(i \in A,i+1 \in A) for given ξ_i > 0. This problem, with random (ξ_i), provides a test example for studying the relationship between optimal and near-optimal solutions of combinatorial optimization problems. We show that, amongst solutions differing from the optimal solution in a small proportion δof places, we can find near-optimal solutions whose objective function value differs from the optimum by a factor of order δ^2 but not smaller order. We conjecture this relationship holds widely in the context of dynamic programming over random data, and Monte Carlo simulations for the Kauffman-Levin NK model are consistent with the conjecture. This work is a technical contribution to a broad program initiated in Aldous-Percus (2003) of relating such scaling exponents to the algorithmic difficulty of optimization problems.

math.PR↗
arXiv2007

Edge Flows in the Complete Random-Lengths Network

Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the $n \to \infty$ limit we find explicitly the empirical distribution of these edge-flows, suitably normalized.

math.PR↗
arXiv2005

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((ξ_i,X_i),i\geq 1). Here (ξ_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 (ξ_i)?

math.PR↗
arXiv2005

Percolation-like Scaling Exponents for Minimal Paths and Trees in the Stochastic Mean Field Model

In the mean field (or random link) model there are $n$ points and inter-point distances are independent random variables. For $0 < \ell < \infty$ and in the $n \to \infty$ limit, let $δ(\ell) = 1/n \times$ (maximum number of steps in a path whose average step-length is $\leq \ell$). The function $δ(\ell)$ is analogous to the percolation function in percolation theory: there is a critical value $\ell_* = e^{-1}$ at which $δ(\cdot)$ becomes non-zero, and (presumably) a scaling exponent $β$ in the sense $δ(\ell) \asymp (\ell - \ell_*)^β$. Recently developed probabilistic methodology (in some sense a rephrasing of the cavity method of Mezard-Parisi) provides a simple albeit non-rigorous way of writing down such functions in terms of solutions of fixed-point equations for probability distributions. Solving numerically gives convincing evidence that $β= 3$. A parallel study with trees instead of paths gives scaling exponent $β= 2$. The new exponents coincide with those found in a different context (comparing optimal and near-optimal solutions of mean-field TSP and MST) and reinforce the suggestion that these scaling exponents determine universality classes for optimization problems on random points.

cond-mat.stat-mech↗
arXiv2004

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 $n$ extant species. The origin of the clade is a random time in the past, whose (improper) distribution is uniform on $(0,\infty)$. 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 $n$ of species at the present time. We study various mathematical properties of this model as $n \to \infty$ 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.

math.PR↗
arXiv2004

A Tractable Complex Network Model based on the Stochastic Mean-field Model of Distance

Much recent research activity has been devoted to empirical study and theoretical models of complex networks (random graphs) with three qualitative features: power-law degree distribution, local clustering of edges, and small diameter. We point out a new (in this context) platform for such models -- the stochastic mean-field model of distance -- and within this platform study a simple two-parameter proportional attachment model. The model is mathematicallly natural, permits a wide variety of explicit calculations, has the desired three qualitative features, and fits the complete range of degree scaling exponents and clustering parameters; in these respects it compares favorably to existing models.

cond-mat.dis-nn↗
arXiv2004

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.

math.PR↗
arXiv2000

The zeta(2) limit in the random assignment problem

The random assignment (or bipartite matching) problem studies the random total cost A_n of the optimal assignment of each of n jobs to each of n machines, where the costs of the n^2 possible job-machine matches has exponential (mean 1) distribution. Mezard - Parisi (1987) used the replica method from statistical physics to argue non-rigorously that EA_n converges to zeta(2) = pi^2/6. Aldous (1992) identified the limit as the optimal solution of a matching problem on an infinite tree. Continuing that approach, we construct the optimal matching on the infinite tree. This yields a rigorous proof of the zeta(2) limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal matching.

math.PR↗