What is... a Graphical Design?

Abstract

Graphical designs are quadrature rules for graphs 𝐺 = (𝑉, 𝐸). Broadly speaking, a graphical design is a relatively small subset of vertices that captures the global behavior of functions 𝑓 ∶ 𝑉 → ℝ. We motivate the precise definition through numerical integration on the sphere.The regular icosahedron (Figure 1) inscribed in the unit sphere 𝕊 2 ⊂ ℝ 3 has the following amazing property: for any polynomial 𝑝(𝑥, 𝑦, 𝑧) of degree at most 5, the average of 𝑝 on the vertices of the icosahedron is equal to the average of 𝑝 over 𝕊 2 . By exploiting symmetry, these 12 points exactly average the 36-dimensional vector space of polynomials with degree at most 5. More generally, a spherical 𝑡-design [3] is a finite subset of points 𝑋 ⊂ 𝕊 2 chosen so that for any polynomial 𝑝(𝑥, 𝑦, 𝑧) of degree at most 𝑡,