Neighbor selection and hitting probability in small-world graphs

Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself. In an attempt to understand how such graphs arise we introduce a different criterion for graphs to be navigable in this sense, relating the neighbor selection of a vertex to the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be shown to lead to graphs with the desired properties. It also leads directly to an evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.Comment: Published in at http://dx.doi.org/10.1214/07-AAP499 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Directed intermittent search for hidden targets

We develop and analyze a stochastic model of directed intermittent search for a hidden target on a one-dimensional track. A particle injected at one end of the track randomly switches between a stationary search phase and a mobile, non-search phase that is biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track or due to competition with other targets. Such a scenario is exemplified by the motor-driven transport of mRNA granules to synaptic targets along a dendrite. We first calculate the hitting probability and conditional mean first passage time (MFPT) for finding a single target. We show that an optimal search strategy does not exist, although for a fixed hitting probability, a unidirectional rather than a partially biased search strategy generates a smaller MFPT. We then extend our analysis to the case of multiple targets, and determine how the hitting probability and MFPT depend on the number of targets

Large deviation estimates of the crossing probability for pinned Gaussian processes

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in $n$ fixed points at $n$ fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motion, $m$-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time.Comment: 33 pages. Keywords: conditioned Gaussian processes; reproducing kernel Hilbert spaces; large deviations; exit time probabilities; Monte Carlo method

On the Moving Boundary Hitting Probability for the Brownian Motion

2000 Mathematics Subject Classification: 60J65.Consider the probability that the Brownian motion hits a moving two-sided boundary by a certain moment. In some special cases we find formulae for this probability

On distributions of functionals of anomalous diffusion paths

Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Levy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrodinger equations that recently appeared in the literature.Comment: 25 pages, 4 figure