Vertex Nomination Between Graphs via Spectral Embedding and Quadratic Programming. Issue 4 (2nd October 2022)

• Record Type: Journal Article
• Authors: Zheng, Runbing; Lyzinski, Vince; Priebe, Carey E.; Tang, Minh
• Is Part Of: Journal of computational and graphical statistics. Volume 31:Issue 4(2022)
• Page Start: 1254
• Page End: 1268
• Publication Date: 2022-10-02
• Subjects: Correlated graphs -- Generalized random dot product graphs -- Point set registration -- Vertex nomination; Mathematical statistics -- Data processing -- Periodicals; Mathematical statistics -- Graphic methods -- Periodicals; 519.50285
• DOI: 10.1080/10618600.2022.2060238
• Languages: English
• ISSNs: 1061-8600

Abstract

Abstract: Given a network and a subset of interesting vertices whose identities are only partially known, the vertex nomination problem seeks to rank the remaining vertices in such a way that the interesting vertices are ranked at the top of the list. An important variant of this problem is vertex nomination in the multiple graphs setting. Given two graphs G 1, G 2 with common vertices and a vertex of interest x ∈ G 1, we wish to rank the vertices of G 2 such that the vertices most similar to x are ranked at the top of the list. The current article addresses this problem and proposes a method that first applies adjacency spectral graph embedding to embed the graphs into a common Euclidean space, and then solves a penalized linear assignment problem to obtain the nomination lists. Since the spectral embedding of the graphs are only unique up to orthogonal transformations, we present two approaches to eliminate this potential nonidentifiability. One approach is based on orthogonal Procrustes and is applicable when there are enough vertices with known correspondence between the two graphs. Another approach uses adaptive point set registration and is applicable when there are few or no vertices with known correspondence. We show that our nomination scheme leads to accurate nomination under a generative model for pairs of random graphs that are approximately low-rank and possibly with pairwise edge correlations. We illustrate our algorithm's performance through simulation studies on synthetic data as well as analysis of a high-school friendship network and analysis of transition rates between web pages on the Bing search engine. Supplementary materials for this article are available and include R code, data, and an appendix with detailed proofs.