SsAG: Summarization and sparsification of Attributed Graphs

We present SSAG, an efficient and scalable method for computing a lossy graph summary that retains the essential structure of the original graph. SSAG computes a sparse representation (summary) of the input graph and also caters for graphs with node attributes. The summary of a graph $G$ is stored as a graph on supernodes (subset of vertices of $G$) and two supernodes are connected by a weighted superedge. The proposed method constructs a summary graph on $k$ supernodes that minimizes the reconstruction error (difference between the original graph and the graph reconstructed from the summary) and maximum homogeneity with respect to attribute values. We construct the summary by iteratively merging a pair of nodes. We derive a closed-form expression to efficiently compute the reconstruction error after merging a pair and approximate this score in constant time. To reduce the search space for selecting the best pair for merging, we assign a weight to each supernode that closely quantifies the contribution of the node in the score of the pairs containing it. We choose the best pair for merging from a random sample made up of supernodes selected with probability proportional to their weights. With weighted sampling, a logarithmic-sized sample yields a comparable summary based on various quality measures. We propose a sparsification step for the constructed summary to reduce the storage cost to a given target size with a marginal increase in reconstruction error. Empirical evaluation on several real-world graphs and comparison with state-of-the-art methods shows that SSAG is up to $5\times$ faster and generates summaries of comparable quality. We further demonstrate the goodness of SSAG by accurately and efficiently answering the queries related to the graph structure and attribute information using the summary only.