Chris J. Michael
ABSTRACT
In numerous applications, spatiotemporal graphs are studied to exploit the structure of underlying data to characterize, control, or predict behavior. Nodes of these graphs exhibit spatiality, while edge connectivity and weighting may be derived from spatial conditions of the incident nodes. Because of the complexity added by the temporal dimension, these graphs are typically modeled in part as a stochastic processes. Though numerous application-defined spatiotemporal graphs with stochastic parameterizations exist, a general stochastic process for such graphs has not yet been formally defined in the literature. In an effort to move towards generalization, we offer a brief introduction to the Stochastic SpatioTemporal (SST) graph model, which describes a graph as a set of initially observed nodes and several sets of stochastic processes: one describing node motion, one describing edge connectivity, and one describing edge weight variation. We propose a Monte Carlo method framework by which temporal graph algorithms that yield numerical or set-based results may be studied for conditions of stability. We demonstrate such a framework by way of a geometric SST graph, which is defined based on the geometric random graph exhibiting Brownian motion of nodes. We offer results to show the points at which node movement and edge weight variation cause the geometric SST graph to become unstable for predicting least-cost paths. Finally, we discuss ongoing research projects and plans currently being undertaken to study and utilize stochastic properties of spatiotemporal network data.
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