Vertex Nomination, Consistent Estimation, and Adversarial Modification
This work addresses graph matching challenges in network analysis, offering incremental theoretical and practical improvements for applications like social network alignment.
The paper tackles the vertex nomination problem for matching vertices of interest across graphs, deriving Bayes optimality for multiple vertices and defining maximal consistency classes, and demonstrates that adversarial contamination degrades performance while a proposed regularization method mitigates this effect in real and synthetic data.
Given a pair of graphs $G_1$ and $G_2$ and a vertex set of interest in $G_1$, the vertex nomination (VN) problem seeks to find the corresponding vertices of interest in $G_2$ (if they exist) and produce a rank list of the vertices in $G_2$, with the corresponding vertices of interest in $G_2$ concentrating, ideally, at the top of the rank list. In this paper, we define and derive the analogue of Bayes optimality for VN with multiple vertices of interest, and we define the notion of maximal consistency classes in vertex nomination. This theory forms the foundation for a novel VN adversarial contamination model, and we demonstrate with real and simulated data that there are VN schemes that perform effectively in the uncontaminated setting, and adversarial network contamination adversely impacts the performance of our VN scheme. We further define a network regularization method for mitigating the impact of the adversarial contamination, and we demonstrate the effectiveness of regularization in both real and synthetic data.