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 multi-graphs setting. Given two graphs $G_1, G_2$ with common vertices and a vertex of interest $x \in G_1$, 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 paper 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 non-identifiability. 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.
翻译:鉴于一个网络和一组有趣的螺旋,其身份仅部分为已知的网络和一组有趣的螺旋,顶点提名问题试图将其余的螺旋排在排序上,使有趣的螺旋排在列表的顶端。这个问题的一个重要变量是多图设置中的顶端提名。鉴于两个图形$_1, G_2$,带有共同的顶端和一个利率的顶端,我们希望将Overticle $x=g_1$,因此,将Overticle 的螺旋排在2$G_2$,这样,在列表的顶端排在最接近 $x$x的顶端。当前纸张将这一问题排在列表的顶端,并提出一种方法,首先将相近的光谱图嵌嵌入多层设置在多面空间中,然后解决一个受惩罚的线性指派问题,以获得提名列表。由于图表的光谱嵌嵌入仅与正向直径直径变,因此我们提出了两种方法可以消除这种潜在的非可识别性。一个方法基于正对等的正读性对等的正对等方法,其基础的直径直对正对等的直路路路系的直路系可能显示,当我们已设定对正对正对等的对等的对等路路路路路路路路路路系的对等的对等的对等的对等路路路路路路路系在可设定的对等路路路路路路路路路路路路路路路路路系的对等路路路路路路由,在可应用的对路由的对路由的对等的对路由的对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路对路