Cubature formulas on combinatorial graphs

TL;DR

This paper develops two types of cubature formulas on combinatorial graphs, enabling exact integration on specific function spaces and with applications in data mining, based on graph Laplacian smoothness.

Contribution

It introduces novel cubature formulas on graphs that are exact on variational splines and bandlimited functions, expanding numerical integration methods in graph analysis.

Findings

Formulas are exact on variational spline spaces.

Formulas are

Accuracy depends on smoothness measured by the Laplace operator.

Abstract

The goal of the paper is to establish cubature formulas on combinatorial graphs. Two types of cubature formulas are developed. Cubature formulas of the first type are exact on spaces of variational splines on graphs. Since badlimited functions can be obtained as limits of variational splines we obtain cubature formulas which are "essentially" exact on spaces of bandlimited functions. Cubature formulas of the second type are exact on spaces of bandlimited functions. Accuracy of cubature formulas is given in terms of smoothness which is measured by means of combinatorial Laplace operator. The results have potential applications to problems that arise in data mining.