Bootstrap estimators for the tail-index and for the count statistics of graphex processes

Naulet, Zacharie; Roy, Daniel M.; Sharma, Ekansh; Veitch, Victor

doi:10.1214/20-ejs1789

Cited by 10 publications

(10 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let A straightforward adaptation of Theorem 1 in [6] and Proposition 1 in [31] yields If we define Q α,c = i 1 wi1>0 1 θi≤α then Q α,c is a homogeneous Poisson process on R + with intensity (0,∞) K ρ(dw 1 , dw 2 , . .…”

Section: A2 Proof Of Theorems 31 32 and Corollary 321mentioning

confidence: 99%

Sparse networks with core-periphery structure

Naik

Caron

Rousseau

2021

Electron. J. Statist.

View full text Add to dashboard Cite

We propose a statistical model for graphs with a core-periphery structure. We give a precise notion of what it means for a graph to have this structure, based on the sparsity properties of the subgraphs of core and periphery nodes. We present a class of sparse graphs with such properties, and provide methods to simulate from this class, and to perform posterior inference. We demonstrate that our model can detect core-periphery structure in simulated and real-world networks.

show abstract

Section: A2 Proof Of Theorems 31 32 and Corollary 321mentioning

confidence: 99%

Sparse networks with core-periphery structure

Naik

Caron

Rousseau

2021

Electron. J. Statist.

View full text Add to dashboard Cite

show abstract

“…There is also related work on bootstrapping [Snijders and Borgatti, 1999, Thompson et al, 2016, Green and Shalizi, 2017, Levin and Levina, 2019, Lin et al, 2020a, jackknife resampling [Lin et al, 2020b] and subsampling [Bhattacharyya and Bickel, 2015, Lunde and Sarkar, 2019, Naulet et al, 2021 in network analysis. In particular, in Lunde and Sarkar [2019], subsampling schemes are applied to the nonzero eigenvalues of the adjacency matrix under low-rank graphon models.…”

Section: Prior Literaturementioning

confidence: 99%

Estimating Graph Dimension with Cross-validated Eigenvalues

Chen¹,

Roch²,

Rohe³

et al. 2021

Preprint

View full text Add to dashboard Cite

In applied multivariate statistics, estimating the number of latent dimensions or the number of clusters is a fundamental and recurring problem. One common diagnostic is the scree plot, which shows the largest eigenvalues of the data matrix in decreasing order; the user searches for a "gap" or "elbow" in the decaying eigenvalues; unfortunately, these patterns can hide beneath the bias of the sample eigenvalues. This methodological problem is conceptually difficult because, in many situations, there is only enough signal to detect a subset of the k population dimensions/eigenvectors. In this situation, one could argue that the correct choice of k is the number of detectable dimensions. We alleviate these problems with cross-validated eigenvalues. Under a large class of random graph models, without any parametric assumptions, we provide a p-value for each sample eigenvector. It tests the null hypothesis that this sample eigenvector is orthogonal to (i.e., uncorrelated with) the true latent dimensions. This approach naturally adapts to problems where some dimensions are not statistically detectable. In scenarios where all k dimensions can be estimated, we prove that our procedure consistently estimates k. In simulations and a data example, the proposed estimator compares favorably to alternative approaches in both computational and statistical performance.

show abstract

“…Studies involving the Kallenberg exchangeable sparse graphs have been occasionally carried out under the assumption that the univariate graphex marginal µ 1 (x) has a power-law-like tail (regularly varying) as x → ∞; see , Naulet et al [2021]. Instead of imposing a condition on µ 1 , we start by imposing important conditions on the generating graphex function, W , which is the direct input for the generation of the underlying network.…”

Section: Regular Variationmentioning

confidence: 99%

Asymptotic Behavior of Common Connections in Sparse Random Networks

Das

Wang

Dai

2021

Preprint

View full text Add to dashboard Cite

Random network models generated using sparse exchangeable graphs have provided a mechanism to study a wide variety of complex real-life networks. In particular, these models help with investigating power-law properties of degree distributions, number of edges, and other relevant network metrics which support the scale-free structure of networks. Previous work on such graphs imposes a marginal assumption of univariate regular variation (e.g., power-law tail) on the bivariate generating graphex function. In this paper, we study sparse exchangeable graphs generated by graphex functions which are multivariate regularly varying. We also focus on a different metric for our study: the distribution of the number of common vertices (connections) shared by a pair of vertices. The number being high for a fixed pair is an indicator of the original pair of vertices being connected. We find that the distribution of number of common connections are regularly varying as well, where the tail indices of regular variation are governed by the type of graphex function used. Our results are verified on simulated graphs by estimating the relevant tail index parameters.

show abstract

Bootstrap estimators for the tail-index and for the count statistics of graphex processes

Cited by 10 publications

References 43 publications

Sparse networks with core-periphery structure

Sparse networks with core-periphery structure

Estimating Graph Dimension with Cross-validated Eigenvalues

Asymptotic Behavior of Common Connections in Sparse Random Networks

Contact Info

Product

Resources

About