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Consistent latent position estimation and vertex classification for random dot product graphs.

Daniel L Sussman1, Minh Tang, Carey E Priebe

  • 1Johns Hopkins University, Baltimore.

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Summary
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This study introduces a method to estimate hidden positions in random dot product graphs using eigen-decomposition. It enables accurate classification of graph vertices, approaching optimal error rates.

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Area of Science:

  • Network Science
  • Machine Learning
  • Statistical Graph Theory

Background:

  • Random dot product graphs (RDPGs) are a fundamental model for network analysis.
  • Estimating latent positions and classifying vertices are key challenges in network science.
  • Existing methods may struggle with consistency and optimal classification accuracy.

Purpose of the Study:

  • To develop a consistent method for estimating latent positions in RDPGs.
  • To achieve Bayes optimal classification for vertices in RDPGs.
  • To validate the proposed methods on both synthetic and real-world network data.

Main Methods:

  • Utilizing the eigen-decomposition of the graph's adjacency matrix.
  • Assuming latent positions are independently and identically distributed (i.i.d.) from an unknown distribution.
  • Applying the k-nearest neighbors classification rule for vertex classification.

Main Results:

  • Demonstrated consistent estimation of latent positions for RDPGs.
  • Showed classification error converges to Bayes optimality as more labeled data becomes available.
  • Successfully evaluated the methods on simulated data and a Wikipedia-derived graph.

Conclusions:

  • Eigen-decomposition provides a robust framework for latent position estimation in RDPGs.
  • The proposed classification strategy achieves near-perfect accuracy with sufficient labeled data.
  • The findings have implications for network analysis, community detection, and link prediction.