Donniell Fishkind

Jordan Yoder, Li Chen, Henry Pao et al.

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

Vince Lyzinski, Donniell Fishkind, Marcelo Fiori et al.

Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational difficulty. Although many heuristics have previously been proposed in the literature to approximately solve graph matching, very few have any theoretical support for their performance. A common technique is to relax the discrete problem to a continuous problem, therefore enabling practitioners to bring gradient-descent-type algorithms to bear. We prove that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimal permutation. These theoretical results suggest that initializing the indefinite algorithm with the convex optimum might yield improved practical performance. Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches.