Sampling perspectives on sparse exchangeable graphs (Q2280538)

This paper studies the sampling convergence as a key concept in the graphon theory of dense graph limit. The sampling convergence generalizes the metric convergence and facilitates the notion of sampling a data set from an infinite size population network. Denote by \(e(G)\) the number of non-loop edges in a graph \(G\). A graph sequence \(G_1, G_2,\cdots\) is sampling convergent if for any \(r>0\) the random graph \(\mathrm{smpl}(G_j,r/\sqrt{2e(G_j)})\) induced by \(r/\sqrt{2e(G_j)}\)-sampling of \(G_j\) converge in distribution as \(j\) tends to infinity. It is shown that a graph sequence generated by integrable graphex \(W\) is almost surely sampling convergent to a canonical dilation of \(W\). It is also proved that if a graph-valued stochastic process \((G_s)_{s>0}\) has the property that for all \(p\in(0,1)\) and all \(s>0\), a \(p\)-sampling of \(G_s\) is equal in distribution to \(G_{ps}\), then there is some graphex \(W\) such that \(G_s=G(\Gamma_s)\) for some \((\Gamma_s)_{s>0}\) generated by graphex \(W\).