NUMERICAL INTEGRATION ON GRAPHS: WHERE TO SAMPLE AND HOW TO WEIGH.

Let G = (V,E,w) be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset W ⊂ V of vertices and weights aw such that 1 | V | ∑ v ∈ V f ( v ) ∼ ∑ w ∈ W a w f ( w ) for functions f : V → ℝ that are 'smooth' with respect to the geometry of the graph; here ~ indicates that we want the right-hand side to be as close to the left-hand side as possible. The main application are problems where f is known to vary smoothly over the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem ('the optimal packing of heat balls'). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method.