Finite differences in a small world

Many complex networks in nature exhibit two properties that are seemingly at odds. They are clustered - neighbors of neighbors are very likely to be neighbors - and they are small worlds - any two nodes can typically be connected by a relatively short path. Watts and Strogatz [17] referred to this as the small world phenomenon and proposed a network model that was shown through simulation to capture the two properties. The model incorporates a parameter that interpolates between fully local and fully global regimes. As the parameter is varied the small world property is roused before the clustering property is lost

Spectral reordering of a range-dependent weighted random graph

Reordering under a random graph hypothesis can be regarded as an extension of clustering and fits into the general area of data mining. Here, we consider a generalization of Grindrod's model and show how an existing spectral reordering algorithm that has arisen in a number of areas may be interpreted from a maximum likelihood range-dependent random graph viewpoint. Looked at this way, the spectral algorithm, which uses eigenvector information from the graph Laplacian, is found to be automatically tuned to an exponential edge density. The connection is precise for optimal reorderings, but is weaker when approximate reorderings are computed via relaxation. We illustrate the performance of the spectral algorithm in the weighted random graph context and give experimental evidence that it can be successful for other edge densities. We conclude by applying the algorithm to a data set from the biological literature that describes cortical connectivity in the cat brain

Google PageRank as mean playing time for pinball on the reverse web

It is known that the output from Google's PageRank algorithm may be interpreted as (a) the limiting value of a linear recurrence relation that is motivated by interpreting links as votes of confidence, and (b) the invariant measure of a teleporting random walk that follows links except for occasional uniform jumps. Here, we show that, for a sufficiently frequent jump rate, the PageRank score may also be interpreted as a mean finishing time for a reverse random walk. At a general step this new process either (i) remains at the current page, (ii) moves to a page that points to the current page, or (iii) terminates. The process is analogous to a game of pinball where a ball bounces between pages before eventually dropping down the exit chute. This new interpretation of PageRank gives another view of the principle that highly ranked pages will be those that are linked into by highly ranked pages that have relatively few outgoing links

Stochastic ordinary differential equations in applied and computational mathematics

Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation

Black-Scholes for scientific computing students

Mathematical finance provides a modern, attractive source of examples and case studies for scientific computing classes. In this installment of Education, I will show how educators can use the Nobel Prize-winning Black-Scholes option valuation theory to motivate exercises in Monte Carlo simulation, matrix computation, and numerical methods for partial differential equations (PDEs)

A matrix perturbation view of the small world phenomenon

We use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440-442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean-field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201-3204]

Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations

This is a continuation of the first author's earlier paper [1] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [1] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs

Black-Scholes option valuation for scientific computing students

Mathematical finance forms a modern, attractive source of examples and case studies for classes in scientific computation. I will show here how the Nobel Prize winning Black-Scholes option valuation theory can be used to motivate exercises in Monte Carlo simulation, matrix computation and numerical methods for partial differential equations

Numerical simulation of a linear stochastic oscillator with additive noise

The ability of numerical methods to reproduce long-time features of a linear stochastic oscillator is examined. It is shown that certain, widely-used, methods fail to capture the correct second moment growth rate, whereas a customized extension of the partitioned Euler method behaves well in this respect. It is also shown that the partitioned Euler method inherits an infinite-oscillation property. A weaker oscillation result is proved for a wide class of numerical methods