A framework for community detection in heterogeneous multi-relational networks

There has been a surge of interest in community detection in homogeneous single-relational networks which contain only one type of nodes and edges. However, many real-world systems are naturally described as heterogeneous multi-relational networks which contain multiple types of nodes and edges. In this paper, we propose a new method for detecting communities in such networks. Our method is based on optimizing the composite modularity, which is a new modularity proposed for evaluating partitions of a heterogeneous multi-relational network into communities. Our method is parameter-free, scalable, and suitable for various networks with general structure. We demonstrate that it outperforms the state-of-the-art techniques in detecting pre-planted communities in synthetic networks. Applied to a real-world Digg network, it successfully detects meaningful communities.Comment: 27 pages, 10 figure

A Unified Community Detection, Visualization and Analysis method

Community detection in social graphs has attracted researchers' interest for a long time. With the widespread of social networks on the Internet it has recently become an important research domain. Most contributions focus upon the definition of algorithms for optimizing the so-called modularity function. In the first place interest was limited to unipartite graph inputs and partitioned community outputs. Recently bipartite graphs, directed graphs and overlapping communities have been investigated. Few contributions embrace at the same time the three types of nodes. In this paper we present a method which unifies commmunity detection for the three types of nodes and at the same time merges partitionned and overlapping communities. Moreover results are visualized in such a way that they can be analyzed and semantically interpreted. For validation we experiment this method on well known simple benchmarks. It is then applied to real data in three cases. In two examples of photos sets with tagged people we reveal social networks. A second type of application is of particularly interest. After applying our method to Human Brain Tractography Data provided by a team of neurologists, we produce clusters of white fibers in accordance with other well known clustering methods. Moreover our approach for visualizing overlapping clusters allows better understanding of the results by the neurologist team. These last results open up the possibility of applying community detection methods in other domains such as data analysis with original enhanced performances.Comment: Submitted to Advances in Complex System

Spectral Detection on Sparse Hypergraphs

We consider the problem of the assignment of nodes into communities from a set of hyperedges, where every hyperedge is a noisy observation of the community assignment of the adjacent nodes. We focus in particular on the sparse regime where the number of edges is of the same order as the number of vertices. We propose a spectral method based on a generalization of the non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance on a planted generative model and compare it with other spectral methods and with Bayesian belief propagation (which was conjectured to be asymptotically optimal for this model). We conclude that the proposed spectral method detects communities whenever belief propagation does, while having the important advantages to be simpler, entirely nonparametric, and to be able to learn the rule according to which the hyperedges were generated without prior information.Comment: 8 pages, 5 figure

Consistency of Spectral Hypergraph Partitioning under Planted Partition Model

Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph partitioning. For instance, consistency results of spectral graph partitioning under the stochastic block model are well known. In this paper, we present a planted partition model for sparse random non-uniform hypergraphs that generalizes the stochastic block model. We derive an error bound for a spectral hypergraph partitioning algorithm under this model using matrix concentration inequalities. To the best of our knowledge, this is the first consistency result related to partitioning non-uniform hypergraphs.Comment: 35 pages, 2 figures, 1 tabl

Community Detection in Hypergraphen

Viele Datensätze können als Graphen aufgefasst werden, d.h. als Elemente (Knoten) und binäre Verbindungen zwischen ihnen (Kanten). Unter dem Begriff der "Complex Network Analysis" sammeln sich eine ganze Reihe von Verfahren, die die Untersuchung von Datensätzen allein aufgrund solcher struktureller Eigenschaften erlauben. "Community Detection" als Untergebiet beschäftigt sich mit der Identifikation besonders stark vernetzter Teilgraphen. Über den Nutzen hinaus, den eine Gruppierung verwandter Element direkt mit sich bringt, können derartige Gruppen zu einzelnen Knoten zusammengefasst werden, was einen neuen Graphen von reduzierter Komplexität hervorbringt, der die Makrostruktur des ursprünglichen Graphen unter Umständen besser hervortreten lässt. Fortschritte im Bereich der "Community Detection" verbessern daher auch das Verständnis komplexer Netzwerke im allgemeinen. Nicht jeder Datensatz lässt sich jedoch angemessen mit binären Relationen darstellen - Relationen höherer Ordnung führen zu sog. Hypergraphen. Gegenstand dieser Arbeit ist die Verallgemeinerung von Ansätzen zur "Community Detection" auf derartige Hypergraphen. Im Zentrum der Aufmerksamkeit stehen dabei "Social Bookmarking"-Datensätze, wie sie von Benutzern von "Bookmarking"-Diensten erzeugt werden. Dabei ordnen Benutzer Dokumenten frei gewählte Stichworte, sog. "Tags" zu. Dieses "Tagging" erzeugt, für jede Tag-Zuordnung, eine ternäre Verbindung zwischen Benutzer, Dokument und Tag, was zu Strukturen führt, die 3-partite, 3-uniforme (im folgenden 3,3-, oder allgemeiner k,k-) Hypergraphen genannt werden. Die Frage, der diese Arbeit nachgeht, ist wie diese Strukturen formal angemessen in "Communities" unterteilt werden können, und wie dies das Verständnis dieser Datensätze erleichtert, die potenziell sehr reich an latenten Informationen sind. Zunächst wird eine Verallgemeinerung der verbundenen Komponenten für k,k-Hypergraphen eingeführt. Die normale Definition verbundener Komponenten weist auf den untersuchten Datensätzen, recht uninformativ, alle Elemente einer einzelnen Riesenkomponente zu. Die verallgemeinerten, so genannten hyper-inzidenten verbundenen Komponenten hingegen zeigen auf den "Social Bookmarking"-Datensätzen eine charakteristische Größenverteilung, die jedoch bspw. von Spam-Verhalten zerstört wird - was eine Verbindung zwischen Verhaltensmustern und strukturellen Eigenschaften zeigt, der im folgenden weiter nachgegangen wird. Als nächstes wird das allgemeine Thema der "Community Detection" auf k,k-Hypergraphen eingeführt. Drei Herausforderungen werden definiert, die mit der naiven Anwendung bestehender Verfahren nicht gemeistert werden können. Außerdem werden drei Familien synthetischer Hypergraphen mit "Community"-Strukturen von steigender Komplexität eingeführt, die prototypisch für Situationen stehen, die ein erfolgreicher Detektionsansatz rekonstruieren können sollte. Der zentrale methodische Beitrag dieser Arbeit besteht aus der im folgenden dargestellten Entwicklung eines multipartiten (d.h. für k,k-Hypergraphen geeigneten) Verfahrens zur Erkennung von "Communities". Es basiert auf der Optimierung von Modularität, einem etablierten Verfahrung zur Erkennung von "Communities" auf nicht-partiten, d.h. "normalen" Graphen. Ausgehend vom einfachst möglichen Ansatz wird das Verfahren iterativ verfeinert, um den zuvor definierten sowie neuen, in der Praxis aufgetretenen Herausforderungen zu begegnen. Am Ende steht die Definition der "ausgeglichenen multi-partiten Modularität". Schließlich wird ein interaktives Werkzeug zur Untersuchung der so gewonnenen "Community"-Zuordnungen vorgestellt. Mithilfe dieses Werkzeugs können die Vorteile der zuvor eingeführten Modularität demonstriert werden: So können komplexe Zusammenhänge beobachtet werden, die den einfacheren Verfahren entgehen. Diese Ergebnisse werden von einer stärker quantitativ angelegten Untersuchung bestätigt: Unüberwachte Qualitätsmaße, die bspw. den Kompressionsgrad berücksichtigen, können über eine größere Menge von Beispielen die Vorteile der ausgeglichenen multi-partiten Modularität gegenüber den anderen Verfahren belegen. Zusammenfassend lassen sich die Ergebnisse dieser Arbeit in zwei Bereiche einteilen: Auf der praktischen Seite werden Werkzeuge zur Erforschung von "Social Bookmarking"-Daten bereitgestellt. Demgegenüber stehen theoretische Beiträge, die für Graphen etablierte Konzepte - verbundene Komponenten und "Community Detection" - auf k,k-Hypergraphen übertragen.Many datasets can be interpreted as graphs, i.e. as elements (nodes) and binary relations between them (edges). Under the label of complex network analysis, a vast array of graph-based methods allows the exploration of datasets purely based on such structural properties. Community detection, as a subfield of network analysis, aims to identify well-connected subparts of graphs. While the grouping of related elements is useful in itself, these groups can furthermore be collapsed into single nodes, creating a new graph of reduced complexity which may better reveal the original graph's macrostructure. Therefore, advances in community detection improve the understanding of complex networks in general. However, not every dataset can be modelled properly with binary relations - higher-order relations give rise to so-called hypergraphs. This thesis explores the generalization of community detection approaches to hypergraphs. In the focus of attention are social bookmarking datasets, created by users of online bookmarking services who assign freely chosen keywords, so-called "tags", to documents. This "tagging" creates, for each tag assignment, a ternary connection between the user, the document, and the tag, inducing particular structures called 3-partite, 3-uniform hypergraphs (henceforth called 3,3- or more generally k,k-hypergraphs). The question pursued here is how to decompose these structures in a formally adequate manner, and how this improves the understanding of these rich datasets. First, a generalization of connected components to k,k-hypergraphs is proposed. The standard definition of connected components here rather uninformatively assigns almost all elements to a single giant component. The generalized so-called hyperincident connected components, however, show a characteristic size distribution on the social bookmarking datasets that is disrupted by, e.g., spamming activity - demonstrating a link between behavioural patterns and structural features that is further explored in the following. Next, the general topic of community detection in k,k-hypergraphs is introduced. Three challenges are posited that are not met by the naive application of standard techniques, and three families of synthetic hypergraphs are introduced containing increasingly complex community setups that a successful detection approach must be able to identify. The main methodical contribution of this thesis consists of the following development of a multi-partite (i.e. suitable for k,k-hypergraphs) community detection algorithm. It is based on modularity optimization, a well-established algorithm to detect communities in non-partite, i.e. "normal" graphs. Starting from the simplest approach possible, the method is successively refined to meet the previously defined as well as empirically encountered challenges, culminating in the definition of the "balanced multi-partite modularity". Finally, an interactive tool for exploring the obtained community assignments is introduced. Using this tool, the benefits of balanced multi-partite modularity can be shown: Intricate patters can be observed that are missed by the simpler approaches. These findings are confirmed by a more quantitative examination: Unsupervised quality measures considering, e.g., compression document the advantages of this approach on a larger number of samples. To conclude, the contributions of this thesis are twofold. It provides practical tools for the analysis of social bookmarking data, complemented with theoretical contributions, the generalization of connected components and modularity from graphs to k,k-hypergraphs

Learning Mixed Membership Community Models in Social Tagging Networks through Tensor Methods

Community detection in graphs has been extensively studied both in theory and in applications. However, detecting communities in hypergraphs is more challenging. In this paper, we propose a tensor decomposition approach for guaranteed learning of communities in a special class of hypergraphs modeling social tagging systems or folksonomies. A folksonomy is a tripartite 3-uniform hypergraph consisting of (user, tag, resource) hyperedges. We posit a probabilistic mixed membership community model, and prove that the tensor method consistently learns the communities under efficient sample complexity and separation requirements

固有値分解とテンソル分解を用いた大規模グラフデータ分析に関する研究

筑波大学 (University of Tsukuba)201