A Reconstructing Graphs from Neighborhood Data

Consider a social network and suppose that we are only given the number of common friends between each pair of users. Can we reconstruct the underlying network? Similarly, consider a set of documents and the words that appear in them. If we only know the number of common words for every pair of documents, as well as the number of common documents for every pair of words, can we infer which words appear in which documents? In this paper, we develop a general methodology for answering questions like the ones above. We formalize these questions in what we call the RECONSTRUCT problem: Given information about the common neighbors of nodes in a network, our goal is to reconstruct the hidden binary matrix that indicates the presence or absence of relationships between individual nodes. In fact, we propose two different variants of this problem: one where the number of connections of every node (i.e., the degree of every node) is known and a second one where it is unknown. We call these variants the degree-aware and the degree-oblivious versions of the RECONSTRUCT problem respectively. Our algorithms for both variants exploit the properties of the singular value decomposition of the hidden binary matrix. More specifically, we show that using the available neighborhood information, we can reconstruct the hidden matrix by finding the components of its singular value decomposition and then combining them appropriately. Our extensive experimental study suggests that our methods are able to reconstruct binary matrices of different characteristics with up to 100% accuracy.