Inferring lockstep behavior from connectivity pattern in large graphs

Given multimillion-node graphs such as “who-follows-whom”, “patent-cites-patent”, “user-likes-page” and “actor/director-makes-movie” networks, how can we find unexpected behaviors? When companies operate on the graphs with monetary incentives to sell Twitter “Followers” and Facebook page “Likes”, the graphs show strange connectivity patterns. In this paper, we study a complete graph from a large Twitter-style social network, spanning up to 3.33 billion edges. We report strange deviations from typical patterns like smooth degree distributions. We find that such deviations are often due to “lockstep behavior” that large groups of followers connect to the same groups of followees. Our first contribution is that we study strange patterns on the adjacency matrix and in the spectral subspaces with respect to several flavors of lockstep. We discover that (a) the lockstep behaviors on the graph shape dense “block” in its adjacency matrix and creates “rays” in spectral subspaces, and (b) partially overlapping of the behaviors shape “staircase” in its adjacency matrix and creates “pearls” in spectral subspaces. The second contribution is that we provide a fast algorithm, using the discovery as a guide for practitioners, to detect users who offer the lockstep behaviors in undirected/directed/bipartite graphs. We carry out extensive experiments on both synthetic and real datasets, as well as public datasets from IMDb and US Patent. The results demonstrate the scalability and effectiveness of our proposed algorithm.