On the Consistency of the Likelihood Maximization Vertex Nomination Scheme: Bridging the Gap Between Maximum Likelihood Estimation and Graph Matching

Graphs are a common data modality, useful for modeling complex relationships between objects, with applications spanning fields as varied as biology (Jeong et al., 2001; Bullmore and Sporns, 2009), sociology (Wasserman and Faust, 1994), and computer vision (Foggia et al., 2014; Kandel et al., 2007), to name a few. For example, in neuroscience, vertices may be neurons and edges adjoin pairs of neurons that share a synapse (Bullmore and Sporns, 2009); in social networks, vertices may correspond to people and edges to friendships between them (Carrington et al., 2005; Yang and Leskovec, 2015); in computer vision, vertices may represent pixels in an image and edges may represent spatial proximity or multi-resolution mappings (Kandel et al., 2007). In many useful networks, vertices with similar attributes form densely-connected communities compared to vertices with highly disparate attributes, and uncovering these communities is an important step in understanding the structure of the network. There is an extensive literature devoted to uncovering this community structure in network data, including methods based on maximum modularity (Newman and Girvan, 2004; Newman, 2006b), spectral partitioning algorithms (Luxburg, 2007; Rohe et al., 2011; Sussman et al., 2012; Lyzinski et al., 2014b), and likelihood-based methods (Bickel and Chen, 2009), among others. In the setting of vertex nomination, one community in the network is of particular interest, and the inference task is to order the vertices into a nomination list with those vertices from the community of interest concentrating at the top of the list.