Dense subgraph discovery

Finding dense subgraphs is a fundamental graph-theoretic problem, that lies in the heart of numerous graph-mining applications, ranging from finding communities in social networks [11, 16], to detecting regulatory motifs in DNA [8] and to identifying real-time stories in news [2]. The problem of finding dense subgraphs has been studied extensively in theoretical computer science [1, 5, 6, 7, 10], and recently, due to the relevance of the problem in real-world applications, it has attracted considerable attention in the data-mining community [4, 16, 18, 20, 21]. In this tutorial we aim to provide a comprehensive overview of (i) major algorithmic techniques for finding dense subgraphs in large graphs and (ii) graph mining applications that rely on dense subgraph extraction. We will present fundamental concepts and algorithms that date back to 80's, as well as the latest advances in the area, from theoretical and from practical point-of-view. We will motivate the problem of finding dense subgraphs by discussing how it can be used in real-world applications. We will discuss different density definitions and the complexity of the corresponding optimization problems. We will also present efficient algorithms for different density measures and under different computational models. Specifically, we will focus on scalable streaming, distributed and MapReduce algorithms. Finally we will discuss problem variants, extensions, and will provide pointers for future research directions.