2015/08/23 by Oliver Knill, Knill, Oliver · 4 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Graph theory and applications #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1508.05657
The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple graphs for which every unit sphere is a d-sphere, the zero locus of (f-c) is a (d-1)-graph for all c different from the range of f. If this Sard lemma is inductively applied to an ordered list functions f1,...,fk in which the functions are extended on the level surfaces, the set of critical values (c1,...,ck) for which F-c=0 is not a (d-k)-graph is a finite set. This discrete Sard result allows to construct explicit graphs triangulating a given algebraic set. We also look at a second setup: for a function F from the vertex set to Rk, we give conditions for which the simultaneous discrete algebraic set F=c defined as the set of simplices of dimension in k, k+1,...,n on which all fi change sign, is a (d-k)-graph in the barycentric refinement of G. This maximal rank condition is adapted from the continuum and the graph F=c is a (n-k)-graph. While now, the critical values can have positive measure, we are closer to calculus: for k=2 for example, extrema of functions f under a constraint g=c happen at points, where the gradients of f and g are parallel D f = L D g, the Lagrange equations on the discrete network. As for an application, we illustrate eigenfunctions of geometric graphs and especially the second eigenvector of 3-spheres, which by Courant-Fiedler has exactly two nodal regions. The separating nodal surface of the second eigenfunction f2 belonging to the smallest nonzero eigenvalue always appears to be a 2-sphere in experiments if G is a 3-sphere.