Predicting Signed Edges with $O(n^{1+o(1)} \log{n})$ Queries

Social networks and interactions in social media involve both positive and negative relationships. Signed graphs capture both types of relationships: positive edges correspond to pairs of "friends", and negative edges to pairs of "foes". The {\em edge sign prediction problem}, which aims to predict whether an interaction between a pair of nodes will be positive or negative, is an important graph mining task for which many heuristics have recently been proposed \cite{leskovec2010predicting,leskovec2010signed}. Motivated by social balance theory, we model the edge sign prediction problem as a noisy correlation clustering problem with two clusters. We are allowed to query each pair of nodes whether they belong to the same cluster or not, but the answer to the query is corrupted with some probability $0<q<\frac{1}{2}$. Let $c=\frac{1}{2}-q$ be the gap. We provide an algorithm that recovers the clustering with high probability in the presence of noise for any constant gap $c$ with $O(n^{{1+\tfrac{1}{\log\log{n}}}\log{n})$} queries. Our algorithm uses simple breadth first search as its main algorithmic primitive. Finally, we provide a novel generalization to $k \geq 3$ clusters and prove that our techniques can recover the clustering if the gap is constant in this generalized setting.