Vertex nomination: The canonical sampling and the extended spectral nomination schemes

TL;DR

This paper introduces scalable vertex nomination schemes for stochastic block models, including a Markov chain Monte Carlo approximation of the optimal canonical scheme and an extended spectral scheme with improved clustering, validated through experiments.

Contribution

It proposes the canonical sampling nomination scheme and extended spectral partitioning scheme, enhancing scalability and accuracy in vertex nomination tasks.

Findings

The canonical sampling scheme converges to the optimal canonical scheme with increased sampling.

The extended spectral scheme improves clustering precision over traditional spectral methods.

Experimental results demonstrate the schemes' effectiveness and computational efficiency.

Abstract

Suppose that one particular block in a stochastic block model is of interest, but block labels are only observed for a few of the vertices in the network. Utilizing a graph realized from the model and the observed block labels, the vertex nomination task is to order the vertices with unobserved block labels into a ranked nomination list with the goal of having an abundance of interesting vertices near the top of the list. There are vertex nomination schemes in the literature, including the optimally precise canonical nomination scheme~$\mathcal{L}^C$ and the consistent spectral partitioning nomination scheme~$\mathcal{L}^P$. While the canonical nomination scheme $\mathcal{L}^C$ is provably optimally precise, it is computationally intractable, being impractical to implement even on modestly sized graphs. With this in mind, an approximation of the canonical scheme---denoted the {\it canonical sampling nomination scheme} $\mathcal{L}^{CS}$---is introduced; $\mathcal{L}^{CS}$ relies on a scalable, Markov chain Monte Carlo-based approximation of $\mathcal{L}^{C}$, and converges to $\mathcal{L}^{C}$ as the amount of sampling goes to infinity. The spectral partitioning nomination scheme is also extended to the {\it extended spectral partitioning nomination scheme}, $\mathcal{L}^{EP}$, which introduces a novel semisupervised clustering framework to improve upon the precision of $\mathcal{L}^P$. Real-data and simulation experiments are employed to illustrate the precision of these vertex nomination schemes, as well as their empirical computational complexity. Keywords: vertex nomination, Markov chain Monte Carlo, spectral partitioning, Mclust MSC[2010]: 60J22, 65C40, 62H30, 62H25