Neal Madras

The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one “effectively independent” sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents areυ=0.7496±0.0007 ind=2 andυ= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200⩽N⩽10000 and 200⩽N⩽ 3000, respectively.

Monte-Carlo approximation algorithms for enumeration problems

We develop polynomial time Monte-Carlo algorithms which produce good approximate solutions to enumeration problems for which it is known that the computation of the exact solution is very hard. We start by developing a Monte-Carlo approximation algorithm for the DNF counting problem, which is the problem of counting the number of satisfying truth assignments to a formula in disjunctive normal form. The input to the algorithm is the formula and two parameters e and δ. The algorithm produces an estimate which is between 1 − ϵ and 1 + ϵ times the number of satisfying truth assignments with probability at least 1 − δ. The running time of the algorithm is linear in the length of the formula times 1ϵ2 times ln(1δ). On the other hand, the problem of computing the exact answer for the DNF counting problem is known to be #P-complete, which implies that there is no polynomial time algorithm for the exact solution if P ≠ NP. This paper improves and gives new applications of some of the work previously reported. Variants of an ϵ, δ approximation algorithm for the DNF counting problem have been highly tailored to be especially efficient for the network reliability problems to which they are applied. In this paper the emphasis is on the development and analysis of a much more efficient ϵ, δ approximation algorithm for the DNF counting problem. The running time of the algorithm presented here substantially improves the running time of versions of this algorithm given previously. We give a new application of the algorithm to a problem which is relevant to physical chemistry and statistical physics. The resulting ϵ, δ approximation algorithm is substantially faster than the fastest known deterministic solution for the problem.

Stability of binary exponential backoff

Binary exponential backoff is a randomized protocol for regulating transmissions on a multiple-access broadcast channel. Ethernet, a local-area network, is built upon this protocol. The fundamental theoretical issue is stability: Does the backlog of packets awaiting transmission remain bounded in time, provided the rates of new packet arrivals are small enough? It is assumed n ≥ 2 stations share the channel, each having an infinite buffer where packets accumulate while the station attempts to transmit the first from the buffer. Here, it is established that binary exponential backoff is stable if the sum of the arrival rates is sufficiently small. Detailed results are obtained on which rates lead to stability when n = 2 stations share the channel. In passing, several other results are derived bearing on the efficiency of the conflict resolution process. Simulation results are reported that, in particular, indicate alternative retransmission protocols can significantly improve performance.

CRITICAL EXPONENTS, HYPERSCALING, AND UNIVERSAL AMPLITUDE RATIOS FOR TWO- AND THREE-DIMENSIONAL SELF-AVOIDING WALKS

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2Δ4 −γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2Δ4 −γ. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios /=0.14026±0.00007, /=0.43961±0.00034, and Ψ*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponentΔ1=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy forΔ1. Earlier Monte Carlo estimates ofv, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios /=0.1599±0.0002 and Ψ*=0.2471±0.0003; since Ψ*>0, hyperscaling holds. The approach to Ψ* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2Δ4 −γ for two-dimensional SAWs.

Nonergodicity of Local, Length-Conserving Monte Carlo Algorithms for the Self-Avoiding Walk

It is proved that every dynamic Monte Carlo algorithm for the self-avoiding walk based on a finite repertoire of local,N-conserving elementary moves is nonergodic (hereN is the number of bonds in the walk). Indeed, for largeN, each ergodic class forms an exponentially small fraction of the whole space. This invalidates (at least in principle) the use of the Verdier-Stockmayer algorithm and its generalizations for high-precision Monte Carlo studies of the self-avoiding walk.

When Is Quarantine a Useful Control Strategy for Emerging Infectious Diseases

Abstract The isolation and treatment of symptomatic individuals, coupled with the quarantining of individuals that have a high risk of having been infected, constitute two commonly used epidemic control measures. Although isolation is probably always a desirable public health measure, quarantine is more controversial. Mass quarantine can inflict significant social, psychological, and economic costs without resulting in the detection of many infected individuals. The authors use probabilistic models to determine the conditions under which quarantine is expected to be useful. Results demonstrate that the number of infections averted (per initially infected individual) through the use of quarantine is expected to be very low provided that isolation is effective, but it increases abruptly and at an accelerating rate as the effectiveness of isolation diminishes. When isolation is ineffective, the use of quarantine will be most beneficial when there is significant asymptomatic transmission and if the asymptomatic period is neither very long nor very short.

Monte Carlo generation of self-avoiding walks with fixed endpoints and fixed length

We propose a new class of dynamic Monte Carlo algorithms for generating self-avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension, and prove that these algorithms are ergodic in all cases. We also prove the ergodicity of a variant of the pivot algorithm.

Modeling Stem Cell Development by Retrospective Analysis of Gene Expression Profiles in Single Progenitor-Derived Colonies

The process of development of various cell types is often based on a linear or deterministic paradigm. This is true, for example, for osteoblast development, a process that occurs through the differentiation of a subset of primitive fibroblast progenitors called colony‐forming unit‐osteoblasts (CFU‐Os). CFU‐O differentiation has been subdivided into three stages: proliferation, extracellular matrix development and maturation, and mineralization, with characteristic changes in gene expression at each stage. Few analyses have asked whether CFU‐O differentiation, or indeed stem cell differentiation in general, may follow more complex and nondeterministic paths, a possibility that may underlie the substantial number of discrepancies in published reports of progenitor cell developmental sequences. We analyzed 99 single colonies of osteoblast stem/primitive progenitor cells cultured under identical conditions. The colonies were analyzed by global amplification poly(A) polymerase chain reaction to determine which of nine genes had been expressed. We used the expression profiles to develop a statistically rigorous map of the cell fate decisions that occur during osteoprogenitor differentiation and show that different developmental routes can be taken to achieve the same end point phenotype. These routes appear to involve both developmental “dead ends” (leading to the expression of genes not correlated with osteoblast‐associated genes or the mature osteoblast phenotype) and developmental flexibility (the existence of multiple gene expression routes to the same developmental end point). Our results provide new insight into the biology of primitive progenitor cell differentiation and introduce a powerful new quantitative method for stem cell lineage analysis that should be applicable to a wide variety of stem cell systems.

A Pattern Theorem for Lattice Clusters

We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration (“pattern”) of sites and bonds can occur in large clusters, then for some constantc>0, it occurs at leastcn times in most clusters of sizen. An analogous theorem for self-avoiding walks was proven in 1963 by Kesten [9]. We use the pattern theorem to prove the convergence of limn→∞an+1/an, wherean is the number of clusters of sizen, up to translation. The results also apply to weighted sums, and in particular, we can takean to be the probability that the percolation cluster containing the origin consists of exactlyn sites. Another consequence is strict inequality of connective constants for sublattices and for certain subclasses of clusters.

Branching random walks on trees

Let p(x, y) be the transition probability of an isotropic random walk on a tree, where each site has d [greater-or-equal, slanted]3 neighbors. We define a branching random walk by letting a particle at site x give birth to a new particle at site y at rate [lambda]dp(x, y), jump to y at rate vdp(x, y), and die at rate [delta]. Let [lambda]2 (respectively, [mu]2) be the infimum of [lambda] such that the process starting with one particle has positive probability of surviving forever (respectively, of having a fixed site occupied at arbitrarily large times). We compute [lambda]2 and [mu]2 exactly, proving that [lambda]2