Random Walks on Random Lattices and Their Applications
White, Ryan Tyler
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This work studies a class of continuous-time, multidimensional random walk processes with mutually dependent random step sizes and their exits from hyperrectangles. Fluctuations of the process about the critical boundary are studied extensively by stochastic analysis and operational calculus. Further, information on the process can be ascertained only upon observations occurring according to a delayed renewal process, rather than in real time. Passage times are thus obscured and results are first derived pertaining to the pre-passage and post-passage observations. Two distinct strategies are developed to combat the crudeness of delayed observations in order to derive more refined information about the processes. The first strategy is to introduce intermediate thresholds on some of the coordinates and considers fluctuations about these intermediate boundaries, which can use information observed over time to continually refine the results. The second "time-sensitive" strategy restricts time to a random time interval, e.g. between the pre-passage and post-passage observations, and revives the real-time paths of the process from the delayed time series. This strategy leads to time-dependent probabilistic results, including joint distributions and conditional distributions and probabilities. In all models, probabilistic results (joint probability transforms under operators, marginal transforms, moments, distributions, probabilities) associated with passage times, excess levels, and the likelihood of threshold(s) to be crossed are derived, and shown to be analytically and numerically tractable under a variety of special cases. Results are tested for accuracy via stochastic simulation. The processes are applied to the detection and prediction of losses to vital networks due to hostile attacks and/or benign failures. The accumulation of losses to a network during a series of loss events is modeled by a 2-dimensional process. The first dimension counts the random numbers of nodes (e.g. routers or operational sites) incapacitated by successive attacks. The nodes have random weights associated with their incapacitation (e.g. loss of operational capacity or cost of repair). The second dimension measures the cumulative weight associated with the nodes lost. The exit from a rectangle corresponds to either type of loss surpassing a threshold, and represents the network entering a critical state.
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