Subjective interestingness of subgraph patterns

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, 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

Gibbs sampling subjectively interesting tiles

International audienceThe local pattern mining literature has long struggled with the so-called pattern explosion problem: the size of the set of patterns found exceeds the size of the original data. This causes computational problems (enumerating a large set of patterns will inevitably take a substantial amount of time) as well as problems for interpretation and usabil-ity (trawling through a large set of patterns is often impractical). Two complementary research lines aim to address this problem. The first aims to develop better measures of interestingness, in order to reduce the number of uninteresting patterns that are returned [6, 10]. The second aims to avoid an exhaustive enumeration of all 'interesting' patterns (where interestingness is quantified in a more traditional way, e.g. frequency), by directly sampling from this set in a way that more 'interest-ing' patterns are sampled with higher probability [2]. Unfortunately, the first research line does not reduce computational cost, while the second may miss out on the most interesting patterns. In this paper, we combine the best of both worlds for mining interesting tiles [8] from binary databases. Specifically, we propose a new pattern sampling approach based on Gibbs sampling, where the probability of sampling a pattern is proportional to their subjective interest-ingness [6]-an interestingness measure reported to better represent true interestingness. The experimental evaluation confirms the theory, but also reveals an important weakness of the proposed approach which we speculate is shared with any other pattern sampling approach. We thus conclude with a broader discussion of this issue, and a forward look

From Sets of Good Redescriptions to Good Sets of Redescriptions

International audienceRedescription mining aims at finding pairs of queries over data variables that describe roughly the same set of observations. These redescriptions can be used to obtain different views on the same set of entities. So far, redescription mining methods have aimed at listing all redescriptions supported by the data. Such an approach can result in many redundant redescriptions and hinder the user's ability to understand the overall characteristics of the data. In this work, we present an approach to find a good set of redescriptions, instead of finding a set of good redescriptions. That is, we present a way to remove the redundant redescriptions from a given set of redescriptions. We measure the redundancy using a framework inspired by the subjective interestingness based on maximum-entropy distributions as proposed by De Bie in 2011. Redescriptions, however, raise their unique requirements on the framework, and our solution differs significantly from the existing ones. Notably, our approach can handle disjunctions and conjunctions in the queries, whereas the existing approaches are limited only to conjunctive queries. The framework also reduces the redundancy in the redescription mining results, as we show in our empirical evaluation

The Minimum Description Length Principle for Pattern Mining: A Survey

This is about the Minimum Description Length (MDL) principle applied to pattern mining. The length of this description is kept to the minimum. Mining patterns is a core task in data analysis and, beyond issues of efficient enumeration, the selection of patterns constitutes a major challenge. The MDL principle, a model selection method grounded in information theory, has been applied to pattern mining with the aim to obtain compact high-quality sets of patterns. After giving an outline of relevant concepts from information theory and coding, as well as of work on the theory behind the MDL and similar principles, we review MDL-based methods for mining various types of data and patterns. Finally, we open a discussion on some issues regarding these methods, and highlight currently active related data analysis problems

Online summarization of dynamic graphs using subjective interestingness for sequential data

Algorithms and the Foundations of Software technolog

The Efficient Discovery of Interesting Closed Pattern Collections

Enumerating closed sets that are frequent in a given database is a fundamental data mining technique that is used, e.g., in the context of market basket analysis, fraud detection, or Web personalization. There are two complementing reasons for the importance of closed sets---one semantical and one algorithmic: closed sets provide a condensed basis for non-redundant collections of interesting local patterns, and they can be enumerated efficiently. For many databases, however, even the closed set collection can be way too large for further usage and correspondingly its computation time can be infeasibly long. In such cases, it is inevitable to focus on smaller collections of closed sets, and it is essential that these collections retain both: controlled semantics reflecting some notion of interestingness as well as efficient enumerability. This thesis discusses three different approaches to achieve this: constraint-based closed set extraction, pruning by quantifying the degree or strength of closedness, and controlled random generation of closed sets instead of exhaustive enumeration. For the original closed set family, efficient enumerability results from the fact that there is an inducing efficiently computable closure operator and that its fixpoints can be enumerated by an amortized polynomial number of closure computations. Perhaps surprisingly, it turns out that this connection does not generally hold for other constraint combinations, as the restricted domains induced by additional constraints can cause two things to happen: the fixpoints of the closure operator cannot be enumerated efficiently or an inducing closure operator does not even exist. This thesis gives, for the first time, a formal axiomatic characterization of constraint classes that allow to efficiently enumerate fixpoints of arbitrary closure operators as well as of constraint classes that guarantee the existence of a closure operator inducing the closed sets. As a complementary approach, the thesis generalizes the notion of closedness by quantifying its strength, i.e., the difference in supporting database records between a closed set and all its supersets. This gives rise to a measure of interestingness that is able to select long and thus particularly informative closed sets that are robust against noise and dynamic changes. Moreover, this measure is algorithmically sound because all closed sets with a minimum strength again form a closure system that can be enumerated efficiently and that directly ties into the results on constraint-based closed sets. In fact both approaches can easily be combined. In some applications, however, the resulting set of constrained closed sets is still intractably large or it is too difficult to find meaningful hard constraints at all (including values for their parameters). Therefore, the last part of this thesis presents an alternative algorithmic paradigm to the extraction of closed sets: instead of exhaustively listing a potentially exponential number of sets, randomly generate exactly the desired amount of them. By using the Markov chain Monte Carlo method, this generation can be performed according to any desired probability distribution that favors interesting patterns. This novel randomized approach complements traditional enumeration techniques (including those mentioned above): On the one hand, it is only applicable in scenarios that do not require deterministic guarantees for the output such as exploratory data analysis or global model construction. On the other hand, random closed set generation provides complete control over the number as well as the distribution of the produced sets.Das Aufzählen abgeschlossener Mengen (closed sets), die häufig in einer gegebenen Datenbank vorkommen, ist eine algorithmische Grundaufgabe im Data Mining, die z.B. in Warenkorbanalyse, Betrugserkennung oder Web-Personalisierung auftritt. Die Wichtigkeit abgeschlossener Mengen ist semantisch als auch algorithmisch begründet: Sie bilden eine nicht-redundante Basis zur Erzeugung von lokalen Mustern und können gleichzeitig effizient aufgezählt werden. Allerdings kann die Anzahl aller abgeschlossenen Mengen, und damit ihre Auflistungszeit, das Maß des effektiv handhabbaren oft deutlich übersteigen. In diesem Fall ist es unvermeidlich, kleinere Ausgabefamilien zu betrachten, und es ist essenziell, dass dabei beide o.g. Eigenschaften erhalten bleiben: eine kontrollierte Semantik im Sinne eines passenden Interessantheitsbegriffes sowie effiziente Aufzählbarkeit. Diese Arbeit stellt dazu drei Ansätze vor: das Einführen zusätzlicher Constraints, die Quantifizierung der Abgeschlossenheit und die kontrollierte zufällige Erzeugung einzelner Mengen anstelle von vollständiger Aufzählung. Die effiziente Aufzählbarkeit der ursprünglichen Familie abgeschlossener Mengen rührt daher, dass sie durch einen effizient berechenbaren Abschlussoperator erzeugt wird und dass desweiteren dessen Fixpunkte durch eine amortisiert polynomiell beschränkte Anzahl von Abschlussberechnungen aufgezählt werden können. Wie sich herausstellt ist dieser Zusammenhang im Allgemeinen nicht mehr gegeben, wenn die Funktionsdomäne durch Constraints einschränkt wird, d.h., dass die effiziente Aufzählung der Fixpunkte nicht mehr möglich ist oder ein erzeugender Abschlussoperator unter Umständen gar nicht existiert. Diese Arbeit gibt erstmalig eine axiomatische Charakterisierung von Constraint-Klassen, die die effiziente Fixpunktaufzählung von beliebigen Abschlussoperatoren erlauben, sowie von Constraint-Klassen, die die Existenz eines erzeugenden Abschlussoperators garantieren. Als ergänzenden Ansatz stellt die Dissertation eine Generalisierung bzw. Quantifizierung des Abgeschlossenheitsbegriffs vor, der auf der Differenz zwischen den Datenbankvorkommen einer Menge zu den Vorkommen all seiner Obermengen basiert. Mengen, die bezüglich dieses Begriffes stark abgeschlossen sind, weisen eine bestimmte Robustheit gegen Veränderungen der Eingabedaten auf. Desweiteren wird die gewünschte effiziente Aufzählbarkeit wiederum durch die Existenz eines effizient berechenbaren erzeugenden Abschlussoperators sichergestellt. Zusätzlich zu dieser algorithmischen Parallele zum Constraint-basierten Vorgehen, können beide Ansätze auch inhaltlich kombiniert werden. In manchen Anwendungen ist die Familie der abgeschlossenen Mengen, zu denen die beiden oben genannten Ansätze führen, allerdings immer noch zu groß bzw. ist es nicht möglich, sinnvolle harte Constraints und zugehörige Parameterwerte zu finden. Daher diskutiert diese Arbeit schließlich noch ein völlig anderes Paradigma zur Erzeugung abgeschlossener Mengen als vollständige Auflistung, nämlich die randomisierte Generierung einer Anzahl von Mengen, die exakt den gewünschten Vorgaben entspricht. Durch den Einsatz der Markov-Ketten-Monte-Carlo-Methode ist es möglich die Verteilung dieser Zufallserzeugung so zu steuern, dass das Ziehen interessanter Mengen begünstigt wird. Dieser neue Ansatz bildet eine sinnvolle Ergänzung zu herkömmlichen Techniken (einschließlich der oben genannten): Er ist zwar nur anwendbar, wenn keine deterministischen Garantien erforderlich sind, erlaubt aber andererseits eine vollständige Kontrolle über Anzahl und Verteilung der produzierten Mengen