Fully Dynamic Algorithm for Top-kk Densest Subgraphs

Given a large graph, the densest-subgraph problem asks to find a subgraph with maximum average degree. When considering the top-$k$ version of this problem, a na\"ive solution is to iteratively find the densest subgraph and remove it in each iteration. However, such a solution is impractical due to high processing cost. The problem is further complicated when dealing with dynamic graphs, since adding or removing an edge requires re-running the algorithm. In this paper, we study the top-$k$ densest-subgraph problem in the sliding-window model and propose an efficient fully-dynamic algorithm. The input of our algorithm consists of an edge stream, and the goal is to find the node-disjoint subgraphs that maximize the sum of their densities. In contrast to existing state-of-the-art solutions that require iterating over the entire graph upon any update, our algorithm profits from the observation that updates only affect a limited region of the graph. Therefore, the top-$k$ densest subgraphs are maintained by only applying local updates. We provide a theoretical analysis of the proposed algorithm and show empirically that the algorithm often generates denser subgraphs than state-of-the-art competitors. Experiments show an improvement in efficiency of up to five orders of magnitude compared to state-of-the-art solutions.Comment: 10 pages, 8 figures, accepted at CIKM 201

Robust Densest Subgraph Discovery

Dense subgraph discovery is an important primitive in graph mining, which has a wide variety of applications in diverse domains. In the densest subgraph problem, given an undirected graph $G=(V,E)$ with an edge-weight vector $w=(w_e)_{e\in E}$, we aim to find $S\subseteq V$ that maximizes the density, i.e., $w(S)/|S|$, where $w(S)$ is the sum of the weights of the edges in the subgraph induced by $S$. Although the densest subgraph problem is one of the most well-studied optimization problems for dense subgraph discovery, there is an implicit strong assumption; it is assumed that the weights of all the edges are known exactly as input. In real-world applications, there are often cases where we have only uncertain information of the edge weights. In this study, we provide a framework for dense subgraph discovery under the uncertainty of edge weights. Specifically, we address such an uncertainty issue using the theory of robust optimization. First, we formulate our fundamental problem, the robust densest subgraph problem, and present a simple algorithm. We then formulate the robust densest subgraph problem with sampling oracle that models dense subgraph discovery using an edge-weight sampling oracle, and present an algorithm with a strong theoretical performance guarantee. Computational experiments using both synthetic graphs and popular real-world graphs demonstrate the effectiveness of our proposed algorithms.Comment: 10 pages; Accepted to ICDM 201

Core Decomposition in Multilayer Networks: Theory, Algorithms, and Applications

Multilayer networks are a powerful paradigm to model complex systems, where multiple relations occur between the same entities. Despite the keen interest in a variety of tasks, algorithms, and analyses in this type of network, the problem of extracting dense subgraphs has remained largely unexplored so far. In this work we study the problem of core decomposition of a multilayer network. The multilayer context is much challenging as no total order exists among multilayer cores; rather, they form a lattice whose size is exponential in the number of layers. In this setting we devise three algorithms which differ in the way they visit the core lattice and in their pruning techniques. We then move a step forward and study the problem of extracting the inner-most (also known as maximal) cores, i.e., the cores that are not dominated by any other core in terms of their core index in all the layers. Inner-most cores are typically orders of magnitude less than all the cores. Motivated by this, we devise an algorithm that effectively exploits the maximality property and extracts inner-most cores directly, without first computing a complete decomposition. Finally, we showcase the multilayer core-decomposition tool in a variety of scenarios and problems. We start by considering the problem of densest-subgraph extraction in multilayer networks. We introduce a definition of multilayer densest subgraph that trades-off between high density and number of layers in which the high density holds, and exploit multilayer core decomposition to approximate this problem with quality guarantees. As further applications, we show how to utilize multilayer core decomposition to speed-up the extraction of frequent cross-graph quasi-cliques and to generalize the community-search problem to the multilayer setting

Incremental and parallel algorithms for dense subgraph mining

The task of maintaining densely connected subgraphs from a continuously evolving graph is important because it solves many practical problems that require constant monitoring over the continuous stream of linked data often represented as a graph. For example, continuous maintenance of a certain group of closely connected nodes can reveal unusual activity over the transaction network, identification, and evolution of active groups in the social network, etc. On the other hand, mining these structures from graph data is often expensive because of the complexity of the computation and the volume of the structures (the number of densely connected structures can be of exponential order on the number of vertices in the graph). One way to deal with the expensive computations is to consider parallel computation. In this thesis, we advance the state of the art by developing provably efficient algorithms for mining maximal cliques and maximal bicliques; two fundamental dense structures. First, we consider the design of efficient algorithms for the maintenance of maximal cliques and maximal bicliques in an evolving network. We observe that it is important to locate the region of the graph in the event of the update so that we can maintain the structures by computing the changes exactly where it is located. Following this observation, we design efficient techniques that find appropriate subgraphs for identifying the changes in the structures. We prove that our algorithms can maintain dense structures efficiently. More specifically, we show that our algorithms can quickly compute the changes when it is small irrespective of the size of the graph. We empirically evaluate our algorithms and show that our algorithms significantly outperform the state of the art algorithms. Next, we consider parallel computation for efficient utilization of the multiple cores in a multi-core computing system so that the expensive mining tasks can be eased off and we can achieve better speedup than their efficient sequential counterparts. We design shared memory parallel algorithms for the mining of maximal cliques and maximal bicliques and we prove the efficiency of the parallel algorithms through showing that the total work performed by the parallel algorithm is equivalent to the time complexity of the best sequential algorithm for doing the same task. Our experimental study shows that we achieve good speedup over the prior state of the art parallel algorithms and significant speedup over the state of the art sequential algorithms. We also show that our parallel algorithms scale almost linearly with the increase in the processor cores