Sampling perspectives on sparse exchangeable graphs

Borgs, Christian; Chayes, Jennifer; Cohn, Henry; Veitch, Victor

doi:10.1214/18-aop1320

Cited by 23 publications

(75 citation statements)

References 26 publications

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“…Here we are most interested in what [25] introduces as GP-convergence, where GP stands for graphex process. This notion is closely related to the notion of sampling convergence for graphs introduced in [4]; see Lemma 5.4 in that paper, as well as the discussion at the end of this section. Janson showed that the following are equivalent if W, W 1 , W 2 , .…”

Section: Definitions and Statements Of Main Resultsmentioning

confidence: 86%

“…Remark 2.3. (1) Following [4], we defined a graphex process as a stochastic process taking values in a space of graphs with labels in R + . Alternatively, one might want to define a graphex process as a process taking values in the space of unlabeled graphs without isolated vertices.…”

Section: Definitions and Statements Of Main Resultsmentioning

confidence: 99%

“…It is instructive to compare our notions of convergence to the notions of graph convergence introduced in [3] and [4]. Before defining these notions, we first introduce the notion of a dilated empirical graphon corresponding to a finite graph G. It involves a "dilation parameter" ρ ∈ R + and is defined as the graphex W(G, ρ) consisting of a zero dust part, a zero star part, a measure space consisting of the vertex set V (G) where each vertex has measure ρ, and a graphon W (G, ρ) which is simply the adjacency matrix of G. The usual way to embed graphs into the space of graphons in the dense case corresponds to ρ = 1/|V (G)|.…”

Section: Definitions and Statements Of Main Resultsmentioning

confidence: 99%

“…The theory of graph limits has been extensively developed for dense graph sequences [7,21,22,8,10,9], but the sparse case is not as well understood. In this paper, we study a model introduced and studied in a sequence of papers [11,26,3,17,25,18,4] based on the notion of graphexes. In contrast to the graphons of the dense theory, which are symmetric two-variable functions defined over a probability space, graphexes are defined over σ-finite measure spaces, and, in addition to a graphon part W , contain two other components: a function S taking values in R + , and a parameter I ∈ R + .…”

mentioning

confidence: 99%

“…We note that in this paper we treat graphexes slightly differently from the definition in [26,17,25,18,4]. Namely, as in [3], we follow the convention from the theory of dense graph limits, and define the graphex process corresponding to a graphex as a process of graphs without loops.…”

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confidence: 99%

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Identifiability for Graphexes and the Weak Kernel Metric

Borgs

Chayes

Cohn

et al. 2019

Bolyai Society Mathematical Studies

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In two recent papers by Veitch and Roy and by Borgs, Chayes, Cohn, and Holden, a new class of sparse random graph processes based on the concept of graphexes over σ-finite measure spaces has been introduced. In this paper, we introduce a metric for graphexes that generalizes the cut metric for the graphons of the dense theory of graph convergence. We show that a sequence of graphexes converges in this metric if and only if the sequence of graph processes generated by the graphexes converges in distribution. In the course of the proof, we establish a regularity lemma and determine which sets of graphexes are precompact under our metric. Finally, we establish an identifiability theorem, characterizing when two graphexes are equivalent in the sense that they lead to the same process of random graphs. Contents 106References 108 1 arXiv:1804.03277v1 [math.PR] 9 Apr 20181 To see that this setting is indeed more general than the assumption of bounded marginals for (unsigned) graphexes we recall that by Proposition 2.4, a graphex with bounded marginals is integrable. Using this, and the fact that by definition, the graphon part of a graphex is bounded, the claim is easy to verify.

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Section: Definitions and Statements Of Main Resultsmentioning

confidence: 86%

Section: Definitions and Statements Of Main Resultsmentioning

confidence: 99%

Section: Definitions and Statements Of Main Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Identifiability for Graphexes and the Weak Kernel Metric

Borgs

Chayes

Cohn

et al. 2019

Bolyai Society Mathematical Studies

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Approximating sparse graphs: The random overlapping communities model

Petti

Vempala

2022

Random Struct Algorithms

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How can we approximate sparse graphs and sequences of sparse graphs (with unbounded average degree)? We consider convergence in the first k moments of the graph spectrum (equivalent to the numbers of closed k-walks) appropriately normalized. We introduce a simple random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The random overlapping communities (ROC) model is specified by a distribution on pairs (s, q), s ∈ Z + , q ∈ (0, 1]. A graph on n vertices with average degree 𝑑 is generated by repeatedly picking pairs (s, q) from the distribution, adding an Erdős-Rényi random graph of edge density q on a subset of vertices chosen by including each vertex with probability s∕n, and repeating this process so that the expected degree is 𝑑. We also show that ROC graphs exhibit an inverse relationship between degree and clustering coefficient, a characteristic of many real-world networks.

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Next waves in veridical network embedding*

Ward

Huang

Davison

et al. 2020

Statistical Analysis

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Embedding nodes of a large network into a metric (e.g., Euclidean) space has become an area of active research in statistical machine learning, which has found applications in natural and social sciences. Generally, a representation of a network object is learned in a Euclidean geometry and is then used for subsequent tasks regarding the nodes and/or edges of the network, such as community detection, node classification and link prediction. Network embedding algorithms have been proposed in multiple disciplines, often with domain-specific notations and details. In addition, different measures and tools have been adopted to evaluate and compare the methods proposed under different settings, often dependent of the downstream tasks. As a result, it is challenging to study these algorithms in the literature systematically. Motivated by the recently proposed PCS framework for Veridical Data Science, we propose a framework for network embedding algorithms and discuss how the principles of predictability, computability, and stability (PCS) apply in this context. The utilization of this framework in network embedding holds the potential to motivate and point to new directions for future research.

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Sampling perspectives on sparse exchangeable graphs

Cited by 23 publications

References 26 publications

Identifiability for Graphexes and the Weak Kernel Metric

Identifiability for Graphexes and the Weak Kernel Metric

Approximating sparse graphs: The random overlapping communities model

Next waves in veridical network embedding*

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