A low discrepancy sequence on graphs
This addresses sampling efficiency for applications like election forecasting and graph-based machine learning, though it appears incremental as an adaptation of existing mathematical concepts to graphs.
The paper tackles the problem of approximating expected values of functions on graph vertices by constructing a sampling scheme analogous to Leja points from complex potential theory, which provides low discrepancy estimates independent of graph size.
Many applications such as election forecasting, environmental monitoring, health policy, and graph based machine learning require taking expectation of functions defined on the vertices of a graph. We describe a construction of a sampling scheme analogous to the so called Leja points in complex potential theory that can be proved to give low discrepancy estimates for the approximation of the expected value by the impirical expected value based on these points. In contrast to classical potential theory where the kernel is fixed and the equilibrium distribution depends upon the kernel, we fix a probability distribution and construct a kernel (which represents the graph structure) for which the equilibrium distribution is the given probability distribution. Our estimates do not depend upon the size of the graph.