[Paper Review] Maximum Likelihood Latent Space Embedding of Logistic Random Dot Product Graphs

Luke J. O’Connor, Muriel Médard+1 more|arXiv (Cornell University)|Oct 3, 2015

Face and Expression Recognition30 references3 citations

TL;DR

This paper proposes a maximum likelihood method for latent space embedding in Logistic Random Dot Product Graphs (Logistic RDPG), using a spectral approach with a novel regression scaling step to achieve asymptotically exact inference. The method improves robustness and accuracy over heuristic spectral methods, especially in community detection tasks.

ABSTRACT

A latent space model for a family of random graphs assigns real-valued vectors to nodes of the graph such that edge probabilities are determined by latent positions. Latent space models provide a natural statistical framework for graph visualizing and clustering. A latent space model of particular interest is the Random Dot Product Graph (RDPG), which can be fit using an efficient spectral method; however, this method is based on a heuristic that can fail, even in simple cases. Here, we consider a closely related latent space model, the Logistic RDPG, which uses a logistic link function to map from latent positions to edge likelihoods. Over this model, we show that asymptotically exact maximum likelihood inference of latent position vectors can be achieved using an efficient spectral method. Our method involves computing top eigenvectors of a normalized adjacency matrix and scaling eigenvectors using a regression step. The novel regression scaling step is an essential part of the proposed method. In simulations, we show that our proposed method is more accurate and more robust than common practices. We also show the effectiveness of our approach over standard real networks of the karate club and political blogs.

Motivation & Objective

• To address the failure of heuristic spectral methods in latent space embedding of Random Dot Product Graphs (RDPG), particularly in separating communities from high-degree nodes.
• To develop a statistically principled, maximum likelihood-based inference method for latent positions in a closely related model: the Logistic RDPG.
• To show that asymptotically exact maximum likelihood estimation of latent vectors is achievable via an efficient spectral algorithm with a regression scaling step.
• To improve robustness and accuracy in graph clustering and community detection compared to standard spectral and heuristic approaches.

Proposed method

• Proposes the Logistic RDPG model, where edge probabilities are modeled via a logistic link function applied to the dot product of latent vectors.
• Uses the top eigenvectors of the mean-centered adjacency matrix as initial estimates of latent positions.
• Introduces a novel regression scaling step to correct eigenvector scaling, ensuring consistency with maximum likelihood estimation.
• Applies a logistic regression model to estimate eigenvalues that maximize the likelihood, linking the spectral solution to the true latent structure.
• Employs a constrained optimization framework to recover latent vectors by maximizing the log-likelihood under rank and norm constraints.
• Theoretical analysis shows that the solution converges uniformly to the maximum likelihood estimate as the network size grows.

Experimental results

Research questions

• RQ1 Can a spectral method achieve asymptotically exact maximum likelihood inference in the Logistic RDPG model?
• RQ2 How does the proposed method improve robustness and accuracy compared to heuristic spectral methods in community detection?
• RQ3 What role does the regression scaling step play in correcting eigenvector scaling and improving inference quality?
• RQ4 Does the method maintain consistency and convergence to the true latent positions under increasing network size?
• RQ5 How does the method perform on real-world networks such as the karate club and political blogs compared to standard approaches?

Key findings

• The proposed method achieves asymptotically exact maximum likelihood inference of latent position vectors in the Logistic RDPG model.
• The regression scaling step is essential for correcting eigenvector scaling and ensures consistency with the maximum likelihood solution.
• Simulations show the method is more accurate and robust than common heuristic spectral methods, especially in detecting communities.
• The method outperforms standard approaches on real-world networks, including the karate club and political blogs datasets.
• Theoretical analysis confirms that the log-likelihood ratio between the proposed solution and the true maximum likelihood estimate converges uniformly to zero as network size increases.
• The method provides a statistically principled alternative to heuristic spectral clustering, with provable consistency and convergence guarantees.