A k-core of a graph is a maximal connected subgraph in which every vertex is connected to at least k vertices in the subgraph. k-core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-Hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for streaming graph data. In this paper, we propose the first incremental k-core decomposition algorithms for streaming graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum k-core values have to be updated, and efficiently process this subgraph to update the k-core decomposition. Our results show a significant reduction in run-time compared to non-incremental alternatives. We show the efficiency of our algorithms on different types of real and synthetic graphs, at different scales. For a graph of 16 million vertices, we observe speedups reaching a million times, relative to the non-incremental algorithms.

Finding dense substructures in a graph is a fundamental graph mining operation, with applications in bioinformatics, social networks, and visualization to name a few. Yet most standard formulations of this problem (like clique, quasiclique, k-densest subgraph) are NP-hard. Furthermore, the goal is rarely to find the "true optimum", but to identify many (if not all) dense substructures, understand their distribution in the graph, and ideally determine relationships among them. Current dense subgraph finding algorithms usually optimize some objective, and only find a few such subgraphs without providing any structural relations.We define the nucleus decomposition of a graph, which represents the graph as a forest of nuclei. Each nucleus is a subgraph where smaller cliques are present in many larger cliques. The forest of nuclei is a hierarchy by containment, where the edge density increases as we proceed towards leaf nuclei. Sibling nuclei can have limited intersections, which enables discovering overlapping dense subgraphs. With the right parameters, the nucleus decomposition generalizes the classic notions of k-cores and k-truss decompositions.We give provably efficient algorithms for nucleus decompositions, and empirically evaluate their behavior in a variety of real graphs. The tree of nuclei consistently gives a global, hierarchical snapshot of dense substructures, and outputs dense subgraphs of higher quality than other stateof-the-art solutions. Our algorithm can process graphs with tens of millions of edges in less than an hour.

We consider the problem of counting motifs in bipartite affiliation networks, such as author-paper, user-product, and actor-movie relations. We focus on counting the number of occurrences of a "butterfly", a complete 2 × 2 biclique, the simplest cohesive higher-order structure in a bipartite graph. Our main contribution is a suite of randomized algorithms that can quickly approximate the number of butterflies in a graph with a provable guarantee on accuracy. An experimental evaluation on large real-world networks shows that our algorithms return accurate estimates within a few seconds, even for networks with trillions of butterflies and hundreds of millions of edges.

Finding dense bipartite subgraphs and detecting the relations among them is an important problem for a liation networks that arise in a range of domains, such as social network analysis, word-document clustering, the science of science, internet advertising, and bioinformatics. However, most dense subgraph discovery algorithms are designed for classic, unipartite graphs. Subsequently, studies on a liation networks are conducted on the co-occurrence graphs (e.g., co-author and co-purchase) that project the bipartite structure to a unipartite structure by connecting two entities if they share an a liation. Despite their convenience, co-occurrence networks come at a cost of loss of information and an explosion in graph sizes, which limit the quality and the e ciency of solutions. We study the dense subgraph discovery problem on bipartite graphs. We de ne a framework of bipartite subgraphs based on the bu er y motif (2,2biclique) to model the dense regions in a hierarchical structure. We introduce e cient peeling algorithms to nd the dense subgraphs and build relations among them. We can identify denser structures compared to the state-of-the-art algorithms on co-occurrence graphs in real-world data. Our analyses on an author-paper network and a user-product network yield interesting subgraphs and hierarchical relations such as the groups of collaborators in the same institution and spammers that give fake ratings.

A k-core of a graph is a maximal connected subgraph in which every vertex is connected to at least k vertices in the subgraph. k-core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for dynamic graph data. In this paper, we propose a suite of incremental k-core decomposition algorithms for dynamic graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum k-core values have changed and efficiently process this subgraph to update the k-core decomposition. We present incremental algorithms for both insertion and deletion operations, and propose auxiliary vertex state maintenance techniques that can further accelerate these operations. Our results show a significant reduction in