Graph sequences sampled from Robinson graphons

Abstract

The function Γ on the space of graphons, introduced in Chuangpishit et al. (2015), aims to measure the extent to which a graphon w exhibits the Robinson property: for all x<y<z, w(x,z)≤min{w(x,y),w(y,z)}. Robinson graphons form a model for graphs with a natural line embedding so that most edges are local. The function Γ is compatible with the cut-norm ‖⋅‖□, in the sense that graphons close in cut-norm have similar Γ-values. In particular, any graphon close in cut-norm to the set of all Robinson graphons has small Γ-values. Here we show the converse, by proving that every graphon w can be approximated by a Robinson graphon Rw so that ‖w−Rw‖□ is bounded in terms of Γ(w). We then use classical techniques from functional analysis to show that a converging graph sequence {Gn} converges to a Robinson graphon if and only if Γ(Gn)→0. Finally, using probabilistic techniques we show that the rate of convergence of Γ for graph sequences sampled from a Robinson graphon can differ substantially depending on how strongly w exhibits the Robinson property.