Mining patterns from multi-relational data is a problem attracting increasing interest within the data mining community. Traditional data mining approaches are typically developed for single-table databases, and are not directly applicable to multi-relational data. Nevertheless, multi-relational data is a more truthful and therefore often also a more powerful representation of reality. Mining patterns of a suitably expressive syntax directly from this representation, is thus a research problem of great importance. In this paper we introduce a novel approach to mining patterns in multi-relational data. We propose a new syntax for multi-relational patterns as complete connected subsets of database entities. We show how this pattern syntax is generally applicable to multi-relational data, while it reduces to well-known tiles " Geerts et al. (Proceedings of Discovery Science, pp 278-289, 2004)" when the data is a simple binary or attribute-value table. We propose RMiner, a simple yet practically efficient divide and conquer algorithm to mine such patterns which is an instantiation of an algorithmic framework for efficiently enumerating all fixed points of a suitable closure operator "Boley et al. (Theor Comput Sci 411(3):691-700, 2010)". We show how the interestingness of patterns of the proposed syntax can conveniently be quantified using a general framework for quantifying subjective interestingness of patterns "De Bie (Data Min Knowl Discov 23(3):407-446, 2011b)". Finally, we illustrate the usefulness and the general applicability of our approach by discussing results on real-world and synthetic databases

The utility of a dense subgraph in gaining a better understanding of a graph has been formalised in numerous ways, each striking a different balance between approximating actual interestingness and computational efficiency. A difficulty in making this trade-off is that, while computational cost of an algorithm is relatively well-defined, a pattern's interestingness is fundamentally subjective. This means that this latter aspect is often treated only informally or neglected, and instead some form of density is used as a proxy. We resolve this difficulty by formalising what makes a dense subgraph pattern interesting to a given user. Unsurprisingly, the resulting measure is dependent on the prior beliefs of the user about the graph. For concreteness, in this paper we consider two cases: one case where the user only has a belief about the overall density of the graph, and another case where the user has prior beliefs about the degrees of the vertices. Furthermore, we illustrate how the resulting interestingness measure is different from previous proposals. We also propose effective exact and approximate algorithms for mining the most interesting dense subgraph according to the proposed measure. Usefully, the proposed interestingness measure and approach lend themselves well to iterative dense subgraph discovery. Contrary to most existing approaches, Learn (2016) 105:41-75 our method naturally allows subsequently found patterns to be overlapping. The empirical evaluation highlights the properties of the new interestingness measure given different prior belief sets, and our approach's ability to find interesting subgraphs that other methods are unable to find.

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Knowledge discovery methods often discover a large number of patterns. Although this can be considered of interest, it certainly presents considerable challenges too. Indeed, this set of patterns often contains lots of uninteresting patterns that risk overwhelming the data miner. In addition, a single interesting pattern can be discovered in a multitude of tiny variations that for all practical purposes are redundant. These issues are referred to as the pattern explosion problem. They lie at the basis of much recent research attempting to quantify interestingness and redundancy between patterns, with the purpose of filtering down a large pattern set to an interesting and compact subset. Many diverse approaches to interestingness and corresponding interestingness measures (IMs) have been proposed in the literature. Some of them, named objective IMs, define interestingness only based on objective criteria of the pattern and data at hand. Subjective IMs additionally depend on the user's prior knowledge about the dataset. Formalizing unexpectedness is probably the most common approach for defining subjective IMs, where a pattern is deemed unexpected if it contradicts the user's expectations about the dataset. Such subjective IMs based on unexpectedness form the focus of this paper. We categorize measures based on unexpectedness into two major subgroups, namely, syntactical and probabilistic approaches. Based on this distinction, we survey different methods for assessing the unexpectedness of patterns with a special focus on frequent itemsets,

Abstract-Three recent trends aim to make local pattern mining more directly suited for use on data as it presents itself in practice, namely in a multi-relational form and affected by noise. The first of these trends is the generalisation of local pattern syntaxes to approximate, noise-tolerant, variants (notably faulttolerant itemset mining and community detection). The second of these trends is to develop pattern syntaxes that are directly applicable to multi-relational data. The third one is to better quantify the interestingness of and redundancy between such local patterns.In this paper we leverage recent results from these lines of research to introduce a noise-tolerant pattern syntax for multirelational data. We show how enumerating all patterns of this syntax in a given database can be done remarkably efficiently. We contribute a way to quantify the interestingness of these patterns, thus overcoming the pattern explosion problem. And finally, we show the usefulness of the pattern syntax and the scalability of the algorithm by presenting experimental results on real world and synthetic data.

Mining patterns from multi-relational data is a problem attracting increasing interest within the data mining community. Traditional data mining approaches are typically developed for highly simplified types of data, such as an attribute-value table or a binary database, such that those methods are not directly applicable to multi-relational data. Nevertheless, multi-relational data is a more truthful and therefore often also a more powerful representation of reality. Mining patterns of a suitably expressive syntax directly from this representation, is thus a research problem of great importance.In this paper we introduce a novel approach to mining patterns in multi-relational data. We propose a new syntax for multi-relational patterns as complete connected subgraphs in a representation of the database as a K-partite graph. We show how this pattern syntax is generally applicable to multirelational data, while it reduces to well-known tiles [7] when the data is a simple binary or attribute-value table. We propose RMiner, an efficient algorithm to mine such patterns, and we introduce a method for quantifying their interestingness when contrasted with prior information of the data miner. Finally, we illustrate the usefulness of our approach by discussing results on real-world and synthetic databases.

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