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

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 $X$

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

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 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.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 $R$ measuring shortness of routes in a network. We illustrate, via Monte Carlo in part, the trade-off between normalized network length and $R$ 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