Finding dense subgraphs is an important problem in graph mining and has many practical applications. At the same time, while large real-world networks are known to have many communities that are not well-separated, the majority of the existing work focuses on the problem of finding a single densest subgraph. Hence, it is natural to consider the question of finding the top-k densest subgraphs. One major challenge in addressing this question is how to handle overlaps: eliminating overlaps completely is one option, but this may lead to extracting subgraphs not as dense as it would be possible by allowing a limited amount of overlap. Furthermore, overlaps are desirable as in most realworld graphs there are vertices that belong to more than one community, and thus, to more than one densest subgraph. In this paper we study the problem of finding top-k overlapping densest subgraphs, and we present a new approach that improves over the existing techniques, both in theory and practice. First, we reformulate the problem definition in a way that we are able to obtain an algorithm with constant-factor approximation guarantee. Our approach relies on using techniques for solving the max-sum diversification problem, which however, we need to extend in order to make them applicable to our setting. Second, we evaluate our algorithm on a collection of benchmark datasets and show that it convincingly outperforms the previous methods, both in terms of quality and efficiency.

Knowledge discovery from data is an inherently iterative process. That is, what we know about the data greatly determines our expectations, and therefore, what results we would find interesting and/or surprising. Given new knowledge about the data, our expectations will change. Hence, in order to avoid redundant results, knowledge discovery algorithms ideally should follow such an iterative updating procedure. With this in mind, we introduce a well-founded approach for succinctly summarizing data with the most informative itemsets; using a probabilistic maximum entropy model, we iteratively find the itemset that provides us the most novel information—that is, for which the frequency in the data surprises us the most—and in turn we update our model accordingly. As we use the maximum entropy principle to obtain unbiased probabilistic models, and only include those itemsets that are most informative with regard to the current model, the summaries we construct are guaranteed to be both descriptive and nonredundant. The algorithm that we present, called mtv, can either discover the top- k most informative itemsets, or we can employ either the Bayesian Information Criterion (bic) or the Minimum Description Length (mdl) principle to automatically identify the set of itemsets that together summarize the data well. In other words, our method will “tell you what you need to know” about the data. Importantly, it is a one-phase algorithm: rather than picking itemsets from a user-provided candidate set, itemsets and their supports are mined on-the-fly. To further its applicability, we provide an efficient method to compute the maximum entropy distribution using Quick Inclusion-Exclusion. Experiments on our method, using synthetic, benchmark, and real data, show that the discovered summaries are succinct, and correctly identify the key patterns in the data. The models they form attain high likelihoods, and inspection shows that they summarize the data well with increasingly specific, yet nonredundant itemsets.

With the fast growth of smart devices and social networks, \ud a lot of computing systems collect data that record different \ud types of activities. An important computational challenge \ud is to analyze these data, extract patterns, and understand \ud activity trends. We consider the problem of mining activity \ud networks to identify interesting events, such as a big concert \ud or a demonstration in a city, or a trending keyword in a user \ud community in a social network. \ud We define an event to be a subset of nodes in the network \ud that are close to each other and have high activity levels. \ud We formalize the problem of event detection using two \ud graph-theoretic formulations. The first one captures the \ud compactness of an event using the sum of distances among \ud all pairs of the event nodes. We show that this formulation \ud can be mapped to the MaxCut problem, and thus, it can \ud be solved by applying standard semidefinite programming \ud techniques. The second formulation captures compactness \ud using a minimum-distance tree. This formulation leads to \ud the prize-collecting Steiner-tree problem, which we solve by \ud adapting existing approximation algorithms. For the two \ud problems we introduce, we also propose efficient and effective \ud greedy approaches and we prove performance guarantees for \ud one of them. We experiment with the proposed algorithms \ud on real datasets from a public bicycling system and a \ud geolocation-enabled social network dataset collected from \ud twitter. The results show that our methods are able to \ud detect meaningful events

Suppose we are given a large graph in which, by some external process, a handful of nodes are marked. What can we say about these nodes? Are they close together in the graph? or, if segregated, how many groups do they form? We approach this problem by trying to find sets of simple connection pathways between sets of marked nodes.We formalize the problem in terms of the Minimum Description Length principle: a pathway is simple when we need only few bits to tell which edges to follow, such that we visit all nodes in a group. Then, the best partitioning is the one that requires the least number of bits to describe the paths that visit all the marked nodes.We prove that solving this problem is NP-hard, and introduce DOT2DOT, an efficient algorithm for partitioning marked nodes by finding simple pathways between nodes. Experimentation shows that DOT2DOT correctly groups nodes for which good connection paths can be constructed, while separating distant nodes.