Sampling and Estimation for (Sparse) Exchangeable Graphs

On Edge Exchangeable Random Graphs

“…In Section 9 we consider an example with sparse graphs, where we can show that the graphs Gt and G m converge in a suitable sense to a more general type of graphons defined by Veitch and Roy [39] as a symmetric measurable function W : R 2 + → [0, 1], see also [3]. Veitch and Roy [40] defined two notions → GP and → GS of convergence for such general graphons on R + (and the even more general graphexes) based on convergence in distribution of the corresponding random graphs. Given a graphon W on R + , we define, see [9; 3; 39], for each r 0 an unlabelled random graph G r (W ) by taking a Poisson process with intensity r on R + , regarded as a random point set {η i } i 1 ; given a realization of this Poisson process, we let Ḡr (W ) be the graph with vertex set N and an edge ij with probability W (η i , η j ) for every pair (i, j) with i < j; finally we let G r (W ) be the graph obtained by deleting all isolated vertices from Ḡr (W ), and then ignoring labels.…”

“…(We assume that W is such that G r (W ) is a.s. finite, see [39] for precise conditions.) We can now define W n → GP W as meaning [40] and [24]. Furthermore, the random graphs G r (W ) are naturally coupled for different r and form an increasing graph process (G r (W )) r 0 .…”

“…Let (G τ k (W )) k be the sequence of different graphs that occur among G r (W ) for r 0. Then [40] and [24]. Remark 3.2.…”

“…However, that is not always the case, and we even give a chameleon example (Theorem 8.6) that has every graph limit as the limit of some subsequence. We give also a sparse example (Example 9.1) with a power-law degree distribution and convergence to a generalized graphon in the sense of Veitch and Roy [40].…”

Sampling perspectives on sparse exchangeable graphs

“…In the setting of dense graphs, these topics meet in the theory of graphons, which are fundamental in the study of graph limits [BCLSV06; LS06; LS07; BCLSV08; BCLSV12] (see [Lov13] for a review) and provide the foundation for many of the statistical network models in current use [NS01; HRH02; ABFX08; MJG09; LOGR12] (see [OR15] for a review). Motivated by the importance of graphons in the dense graph setting, a recent series of papers [CF14; HSM16; VR15; BCCH16; VR16; TC16; Jan17] has developed a generalization of the graphon framework to the regime of sparse graphs, both as a tool for statistical network modeling [VR15;BCCH16] and estimation [VR16], and as the central element of a limit theory for large graphs [BCCH16] (see also [Jan16]). This generalization is compelling in that it preserves many of the desirable properties of the graphon framework, while simultaneously allowing much greater flexibility.…”