Vertex Nomination, Consistent Estimation, and Adversarial Modification

Given a pair of graphs G1 and G2 and a vertex set of interest in G1, the vertex nomination (VN) problem seeks to find the corresponding vertices of interest in G2 (if they exist) and produce a rank list of the vertices in G2, with the corresponding vertices of interest in G2 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.