In this paper, we revisit pattern mining and study the distribution underlying a binary dataset thanks to the closure structure which is based on passkeys, i.e., minimum generators in equivalence classes robust to noise. We introduce $\Delta$-closedness, a generalization of the closure operator, where $\Delta$ measures how a closed set differs from its upper neighbors in the partial order induced by closure. A $\Delta$-class of equivalence includes minimum and maximum elements and allows us to characterize the distribution underlying the data. Moreover, the set of $\Delta$-classes of equivalence can be partitioned into the so-called $\Delta$-closure structure. In particular, a $\Delta$-class of equivalence with a high level demonstrates correlations among many attributes, which are supported by more observations when $\Delta$ is large. In the experiments, we study the $\Delta$-closure structure of several real-world datasets and show that this structure is very stable for large $\Delta$ and does not substantially depend on the data sampling used for the analysis.

In this paper, we are revisiting pattern mining and especially itemset mining, which allows one to analyze binary datasets in searching for interesting and meaningful association rules and respective itemsets in an unsupervised way. While a summarization of a dataset based on a set of patterns does not provide a general and satisfying view over a dataset, we introduce a concise representation --the closure structure-- based on closed itemsets and their minimum generators, for capturing the intrinsic content of a dataset. The closure structure allows one to understand the topology of the dataset in the whole and the inherent complexity of the data. We propose a formalization of the closure structure in terms of Formal Concept Analysis, which is well adapted to study this data topology. We present and demonstrate theoretical results, and as well, practical results using the GDPM algorithm. GDPM is rather unique in its functionality as it returns a characterization of the topology of a dataset in terms of complexity levels, highlighting the diversity and the distribution of the itemsets. Finally, a series of experiments shows how GDPM can be practically used and what can be expected from the output.