Consistent Latent Position Estimation and Vertex Classification for Random Dot Product Graphs

2014 ◽  
Vol 36 (1) ◽  
pp. 48-57 ◽  
Author(s):  
Daniel L. Sussman ◽  
Minh Tang ◽  
Carey E. Priebe
Keyword(s):  
Position Estimation ◽  
Product Graphs ◽  
Dot Product ◽  
Vertex Classification ◽  
Dot Product Graphs

scholarly journals Directed Random Dot Product Graphs

2008 ◽  
Vol 5 (1-2) ◽  
pp. 91-111 ◽  
Author(s):  
Stephen J. Young ◽  
Edward Scheinerman
Keyword(s):  
Product Graphs ◽  
Dot Product ◽  
Dot Product Graphs

scholarly journals What Graphs are 2-Dot Product Graphs?

2021 ◽  
pp. 1-16
Author(s):  
Matthew Johnson ◽  
Daniël Paulusma ◽  
Erik Jan van Leeuwen
Keyword(s):  
Social Networks ◽  
Product Formula ◽  
Np Hard ◽  
Intersection Graphs ◽  
Graph Classes ◽  
Product Graphs ◽  
Positive Threshold ◽  
Dot Product ◽  
Vector Formula ◽  
Dot Product Graphs

Let [Formula: see text] be an integer. From a set of [Formula: see text]-dimensional vectors, we obtain a [Formula: see text]-dot by letting each vector [Formula: see text] correspond to a vertex [Formula: see text] and by adding an edge between two vertices [Formula: see text] and [Formula: see text] if and only if their dot product [Formula: see text], for some fixed, positive threshold [Formula: see text]. Dot product graphs can be used to model social networks. Recognizing a [Formula: see text]-dot product graph is known to be NP -hard for all fixed [Formula: see text]. To understand the position of [Formula: see text]-dot product graphs in the landscape of graph classes, we consider the case [Formula: see text], and investigate how [Formula: see text]-dot product graphs relate to a number of other known graph classes including a number of well-known classes of intersection graphs.

Change Point Detection in Weighted and Directed Random Dot Product Graphs

2021 ◽  
Author(s):  
Federico Larroca ◽  
Paola Bermolen ◽  
Marcelo Fiori ◽  
Gonzalo Mateos
Keyword(s):  
Change Point ◽  
Change Point Detection ◽  
Product Graphs ◽  
Dot Product ◽  
Point Detection ◽  
Dot Product Graphs

scholarly journals What Graphs are 2-Dot Product Graphs?

2015 ◽  
Vol 49 ◽  
pp. 705-711
Author(s):  
Matthew Johnson ◽  
Erik Jan van Leeuwen ◽  
Daniël Paulusma
Keyword(s):  
Product Graphs ◽  
Dot Product ◽  
Dot Product Graphs

scholarly journals Optimal Bayesian estimation for random dot product graphs

Biometrika ◽  
2020 ◽  
Vol 107 (4) ◽  
pp. 875-889
Author(s):  
Fangzheng Xie ◽  
Yanxun Xu
Keyword(s):  
Lower Bound ◽  
Estimation Method ◽  
Positive Semidefinite ◽  
Contraction Rate ◽  
Convergence Results ◽  
Product Graphs ◽  
Spectral Embedding ◽  
Dot Product ◽  
Probability Matrices ◽  
Dot Product Graphs

Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating the latent positions, and show that posterior spectral embedding achieves this lower bound in the following two senses: it both results in a minimax-optimal posterior contraction rate and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models with positive semidefinite block probability matrices, strengthening an existing result concerning the number of misclustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogue of adjacency spectral embedding, but the resulting posterior contraction rate is suboptimal by an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of Wikipedia graph data.

A Limit Theorem for Scaled Eigenvectors of Random Dot Product Graphs

Sankhya A ◽  
2015 ◽  
Vol 78 (1) ◽  
pp. 1-18 ◽  
Author(s):  
A. Athreya ◽  
C. E. Priebe ◽  
M. Tang ◽  
V. Lyzinski ◽  
D. J. Marchette ◽  
...  
Keyword(s):  
Limit Theorem ◽  
Product Graphs ◽  
Dot Product ◽  
Dot Product Graphs

scholarly journals Algorithms for diversity and clustering in social networks through dot product graphs

2015 ◽  
Vol 41 ◽  
pp. 48-55 ◽  
Author(s):  
Matthew Johnson ◽  
Daniël Paulusma ◽  
Erik Jan van Leeuwen
Keyword(s):  
Social Networks ◽  
Product Graphs ◽  
Dot Product ◽  
Dot Product Graphs
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