The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between two people depends on their attributes, such as their age, address, and hobbies. The connectivity of a graph can thus possibly be understood in terms of patterns of the form 'the subgroup of individuals with properties X are often (or rarely) friends with individuals in another subgroup with properties Y'. Such rules present potentially actionable and generalizable insights into the graph. We present a method that finds pairs of node subgroups between which the edge density is interestingly high or low, using an information-theoretic definition of interestingness. This interestingness is quantified subjectively, to contrast with prior information an analyst may have about the graph. This view immediately enables iterative mining of such patterns. Our work generalizes prior work on dense subgraph mining (i.e. subgraphs induced by a single subgroup). Moreover, not only is the proposed method more general, we also demonstrate considerable practical advantages for the single subgroup special case.

Numerical time series data are pervasive, originating from sources as diverse as wearable devices, medical equipment, to sensors in industrial plants. In many cases, time series contain interesting information in terms of subsequences that recur in approximate form, so-called motifs. Major open challenges in this area include how one can formalize the interestingness of such motifs and how the most interesting ones can be found. We introduce a novel approach that tackles these issues. We formalize the notion of such subsequence patterns in an intuitive manner and present an information-theoretic approach for quantifying their interestingness with respect to any prior expectation a user may have about the time series. The resulting interestingness measure is thus a subjective measure, enabling a user to find motifs that are truly interesting to them. Although finding the best motif appears computationally intractable, we develop relaxations and a branch-and-bound approach implemented in a constraint programming solver. As shown in experiments on synthetic data and two real-world datasets, this enables us to mine interesting patterns in small or mid-sized time series.

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The connectivity structure of graphs is typically related to the attributes of the vertices. In social networks for example, the probability of a friendship between any pair of people depends on a range of attributes, such as their age, residence location, workplace, and hobbies. The high-level structure of a graph can thus possibly be described well by means of patterns of the form ‘the subgroup of all individuals with certain properties X are often (or rarely) friends with individuals in another subgroup defined by properties Y’, ideally relative to their expected connectivity. Such rules present potentially actionable and generalizable insight into the graph. Prior work has already considered the search for dense subgraphs (‘communities’) with homogeneous attributes. The first contribution in this paper is to generalize this type of pattern to densities between a pair of subgroups, as well as between all pairs from a set of subgroups that partition the vertices. Second, we develop a novel information-theoretic approach for quantifying the subjective interestingness of such patterns, by contrasting them with prior information an analyst may have about the graph’s connectivity. We demonstrate empirically that in the special case of dense subgraphs, this approach yields results that are superior to the state-of-the-art. Finally, we propose algorithms for efficiently finding interesting patterns of these different types.