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Register a new userDiscussions of the paper Sparse graphs using exchangeable random measures by F. Caron and E. B. Fox
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Julyan Arbel
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
Authors
Julyan Arbel
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These are written discussions of the paper Sparse graphs using exchangeable random measures by Franc{c}ois Caron and Emily B. Fox, contributed to the Journal of the Royal Statistical Society Series B.
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We introduce two new bootstraps for exchangeable random graphs. One, the empirical graphon, is based purely on resampling, while the other, the histogram stochastic block model, is a model-based sieve bootstrap. We show that both of them accurately approximate the sampling distributions of motif densities, i.e., of the normalized counts of the number of times fixed subgraphs appear in the network. These densities characterize the distribution of (infinite) exchangeable networks. Our bootstraps therefore give, for the first time, a valid quantification of uncertainty in inferences about fundamental network statistics, and so of parameters identifiable from them.
Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay between the statistical modeling of network data, the theory of large graph limits, exchangeability, and network sampling. The purpose of the present paper is to clarify the relationships between these subjects by explaining each in terms of a certain natural sampling scheme associated with the graphex model. The first main technical contribution is the introduction of sampling convergence, a new notion of graph limit that generalizes left convergence so that it becomes meaningful for the sparse graph regime. The second main technical contribution is the demonstration that the (somewhat cryptic) notion of exchangeability underpinning the graphex framework is equivalent to a more natural probabilistic invariance expressed in terms of the sampling scheme.
This note is a collection of several discussions of the paper Beyond subjective and objective in statistics, read by A. Gelman and C. Hennig to the Royal Statistical Society on April 12, 2017, and to appear in the Journal of the Royal Statistical Society, Series A.
Kallenberg (2005) provided a necessary and sufficient condition for the local finiteness of a jointly exchangeable random measure on $R_+^2$. Here we note an additional condition that was missing in Kallenbergs theorem, but was implicitly used in the proof. We also provide a counter-example when the additional condition does not hold.
A star $k$-coloring is a proper $k$-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than $frac{5}{2}$ has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219].
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