arXiv subjects
Publications and source records attributed to A. D. Barbour.
The paper consists of a brief survey of the use of Stein's method in network analysis. Topics covered include normal and Poisson approximation of graph--based statistics, approximating an exponential random graph by a Bernoulli model, and comparison of different random geometric graph models.
In directed random graphs, in which edges can be assigned to have one of two directions, or perhaps both, the distance between two vertices $v$ and $v'$ can be computed along paths that are directed from $v$ to $v'$, or along paths that are directed from $v'$ to $v$. These two distances are in general dependent. Here, we approximate their joint distribution in the setting of the directed Bernoulli random graph $\mathcal{DG}(n,p,θ)$, obtained as a natural extension of the Bernoulli random graph $\mathcal{G}(n,p)$ by assigning directions to the edges independently, bidirectional with probability $θ$, and either of the two possible choices of single direction with probability $\frac12(1-θ)$. The approximation involves two independent copies of a trivariate limiting random vector $(W^*_1,W^*_2,W_3)$ associated with a $3$-type Bienaymé--Galton--Watson process. The approximation error is shown to be typically of order $O(n^{-1/2}\log n)$; this asymptotic order is likely to be optimal, even for the corresponding approximation in the Bernoulli random graph $\mathcal{G}(n,p)$.
Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals $h$ of a càdlàg random process $X$ of interest and the expectations of the same functionals of a well understood target random process $Z$ with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we prove an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded -- examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events -- in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of $X$ that is closely related to classical tightness criteria, a similar expectation for $Z$, and, for the indicator of a set $K$, the probability $\mathbb{P}(Z \in K^θ\setminus K^{-θ})$ that the target process is close to the boundary of $K$.
We develop a general approach to Stein's method for approximating a random process in the path space $D([0,T]\to R^d)$ by a real continuous Gaussian process. We then use the approach in the context of processes that have a representation as integrals with respect to anunderlying point process, deriving a general quantitative Gaussian approximation. The error bound is expressed in terms of couplings of the original process to processes generated from the reduced Palm measures associated with the point process. As applications, we study certain $\text{GI}/\text{GI}/\infty$ queues in the "heavy traffic" regime.
We study the degree distribution of a randomly chosen vertex in a duplication--divergence graph, under a variety of different generalizations of the basic model of Bhan, Galas and Dewey (2002) and Vázquez, Flammini, Maritan and Vespignani (2003). We pay particular attention to what happens when a non-trivial proportion of the vertices have large degrees, establishing a central limit theorem for the logarithm of the degree distribution. Our approach, as in Jordan (2018) and Hermann and Pfaffelhuber (2021), relies heavily on the analysis of related birth--catastrophe processes, and couplings are used to show that a number of different formulations of the process have asymptotically similar expected degree distributions.
The Network Disturbance Model of Doreian (1989) expresses the dependency between observations taken at the vertices of a network by modelling the correlation between neighbouring vertices, using a single correlation parameter $ρ$. It has been observed that estimation of $ρ$ in dense graphs, using the method of Maximum Likelihood, leads to results that can be both biased and very unstable. In this paper, we sketch why this is the case, showing that the variability cannot be avoided, no matter how large the network. We also propose a more intuitive estimator of $ρ$, which shows little bias. The related Network Effects Model is briefly discussed.
We derive explicit central moment inequalities for random variables that admit a Stein coupling, such as exchangeable pairs, size--bias couplings or local dependence, among others. The bounds are in terms of moments (not necessarily central) of variables in the Stein coupling, which are typically local in some sense, and therefore easier to bound. In cases where the Stein couplings have the kind of behaviour leading to good normal approximation, the central moments are closely bounded by those of a normal. We show how the bounds can be used to produce concentration inequalities, and compare them to those existing in related settings. Finally, we illustrate the power of the theory by bounding the central moments of sums of neighbourhood statistics in sparse Erdős--Rényi random graphs.
We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with $d$ neighbours in a germ--grain model, and the number of degree-$d$ vertices in an Erdős--Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.
We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form $T := \sum_{v \in V} H_v$, where $V$ is the vertex set, and $H_v$ depends on a neighbourhood in the graph around $v$ of size at most $\ell$. The error bound is expressed in terms of $\ell$, $|V|$, an almost sure bound on $H_v$, the maximum vertex degree $d_{\max}$ and the variance of $T$. Under suitable assumptions on the convergence of the empirical degree distributions to a limiting distribution, we deduce that the size of the giant component in the configuration model has asymptotically Gaussian fluctuations.
The Aldous gossip process represents the dissemination of information in geographical space as a process of locally deterministic spread, augmented by random long range transmissions. Starting from a single initially informed individual, the proportion of individuals informed follows an almost deterministic path, but for a random time shift, caused by the stochastic behaviour in the very early stages of development. In this paper, it is shown that, even with the extra information available after a substantial development time, this broad description remains accurate to first order. However, the precision of the prediction is now much greater, and the random time shift is shown to have an approximately normal distribution, with mean and variance that can be computed from the current state of the process.
We establish two theorems for assessing the accuracy in total variation of multivariate discrete normal approximation to the distribution of an integer valued random vector $W$. The first is for sums of random vectors whose dependence structure is local. The second applies to random vectors~$W$ resulting from integrating the $\mathbb{Z}^d$-valued marks of a marked point process with respect to its ground process. The error bounds are of magnitude comparable to those given in Rinott and Rotar (1996), but now with respect to the stronger total variation distance. Instead of requiring the summands to be bounded, we make third moment assumptions. We demonstrate the use of the theorems in four applications: monochrome edges in vertex coloured graphs, induced triangles and $2$-stars in random geometric graphs, the times spent in different states by an irreducible and aperiodic finite Markov chain, and the maximal points in different regions of a homogeneous Poisson point process.
We consider the approximation of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset $Ω$ of Euclidean space. The approximation is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at $z$ being occupied is shown to be close to $q_1(z)$, the equilibrium occupation probability in Levins's model, at any point $z \in Ω$ not too close to the boundary, if the local colonization pressure and extinction rates appropriate to $z$ are assumed. The approximation is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.
We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erdős-Rényi random graph, and of the Curie-Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation bounds seem to be new. The general result follows from Stein's method, and requires a new bound on the Stein solution for the Poisson distribution, which is of general interest.
For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein--Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein's method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration--death processes with Poisson distributions as equilibria. The method is illustrated by providing total variation error bounds for the approximation of the equilibrium distribution of one Markov jump process by that of another. In a companion paper, it is shown how to use the method for discrete normal approximation in ${\mathbb Z}^d$.
The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in ${\mathbb Z}^d$. We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.
We consider the motion of a particle on a Galton Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours is determined by a random process. Given the tree, positive weights are assigned to the edges in such a way that, viewed along any line of descent, they evolve as a random process. In order to introduce our method for proving transience or recurrence, we first suppose that the weights are i.i.d., reproving a result of Lyons and Pemantle. We then extend the argument to allow a Markovian environment, and finally to a random walk on a Markovian environment that changes the environment. Our approach involves studying the typical behaviour of processes on fixed lines of descent, which we then show determines the behaviour of the process on the whole tree.
The paper presents a phenomenon occurring in population processes that start near zero and have large carrying capacity. By the classical result of Kurtz~(1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to infinity, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth and death process.
The paper gives the bounds on the solutions to a Stein equation for the negative binomial distribution that are needed for approximation in terms of the Wasserstein metric. The proofs are probabilistic, and follow the approach introduced in Barbour and Xia (Bernoulli 12 (2006) 943-954). The bounds are used to quantify the accuracy of negative binomial approximation to parasite counts in hosts. Since the infectivity of a population can be expected to be proportional to its total parasite burden, the Wasserstein metric is the appropriate choice.