Utilizing covariates in partially observed networks

In many applications of network analysis one observes the network of interest imperfectly. Inference on such networks requires one to either impute the unobserved information, or to be robust to the missing information. In many applications, such as social networks, brain connectomes, communications networks, etc. these graphs can be extremely large, which can make imputation problematic. Typically one has little to no control over the missing information. Additionally, there is often meta-data associated with the network, and incorporating these covariates can improve the inference and provide some level of robustness under the missing data. We will describe a family of spectral graph algorithms that allow one to utilize covariates in a natural way, and will illustrate the methodology on a large graph derived from Twitter, in which one observes most of the tweets containing geographic information (latitude and longitude of the device used to send the tweet) and builds the mentions graph, in which a directed edge indicates that a tweet from one user mentioned another user. Note that since we observe only those tweets with a location, we do not observe tweets from many of the users mentioned, and so do not have information about their outward edges or such covariates as their location, the languages they speak, etc. Note that in the case of the latter, we observe the meta-data partially as well - we observe the language of the tweets mentioning these users, but not those sent by them. We will describe a method to utilize the geographic information of the subset of users for which this is observed to perform inference on the overall graph, such as inferring missing edges and inferring the location of users whose devices do not provide this information.