Random Geometric Complexes and Graphs on Riemannian Manifolds in the Thermodynamic Limit

Abstract

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence results for the counting measure of types of components of each k-skeleton of a random geometric complex. As a consequence, in the case of the 1-skeleton (i.e., for random geometric graphs) we show that the empirical spectral measure associated to the normalized Laplace operator converges to a deterministic measure.

Spectral classes of hypergraphs

“…In Section 3 we define hypergraph spectral measures and spectral classes, and in Section 4 we establish them for some given families of hypergraphs. Finally, in Section 5 we generalize two results proved in [4] on the spectral classes of families of graphs that only differ by a fixed number of edges.…”

“…Gu et al also introduced and investigated spectral classes with the aim of studying the asymptotics of spectra of growing families of graphs. Moreover, in [4], Lerario together with the author of this paper extended this theory and established results on spectral classes that also involve other operators associated to a graph, such as the degree matrix, the adjacency matrix and the Kirchhoff Laplacian.…”

“…Similarly, Thm. 6.4 in [4] states that the difference of the spectral measures of (Γ 1,n ) n and (Γ 2,n ) n goes to zero weakly, w.r.t. A, D, K and L, without the assumption that the corresponding spectral measures have weak limits.…”

“…5.4 and by Prop. 6.7 in [4], for each uniformly continuous function f : R → R and for each ε > 0 there exists δ > 0 such that…”

“…Remark 5.2. As shown in [4], if for n even we let Γ n,1 be the path on n vertices and we let Γ n,2 be the disjoint union of two paths on n/2 vertices, then the total variation distance between the two measures w.r.t. L does not tend to zero as n → ∞.…”