Multi-level algorithms for modularity clustering

Modularity is one of the most widely used quality measures for graph clusterings. Maximizing modularity is NP-hard, and the runtime of exact algorithms is prohibitive for large graphs. A simple and effective class of heuristics coarsens the graph by iteratively merging clusters (starting from singletons), and optionally refines the resulting clustering by iteratively moving individual vertices between clusters. Several heuristics of this type have been proposed in the literature, but little is known about their relative performance. This paper experimentally compares existing and new coarsening- and refinement-based heuristics with respect to their effectiveness (achieved modularity) and efficiency (runtime). Concerning coarsening, it turns out that the most widely used criterion for merging clusters (modularity increase) is outperformed by other simple criteria, and that a recent algorithm by Schuetz and Caflisch is no improvement over simple greedy coarsening for these criteria. Concerning refinement, a new multi-level algorithm is shown to produce significantly better clusterings than conventional single-level algorithms. A comparison with published benchmark results and algorithm implementations shows that combinations of coarsening and multi-level refinement are competitive with the best algorithms in the literature.Comment: 12 pages, 10 figures, see http://www.informatik.tu-cottbus.de/~rrotta/ for downloading the graph clustering softwar

Ensemble clustering via heuristic optimisation

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel UniversityTraditional clustering algorithms have different criteria and biases, and there is no single algorithm that can be the best solution for a wide range of data sets. This problem often presents a significant obstacle to analysts in revealing meaningful information buried among the huge amount of data. Ensemble Clustering has been proposed as a way to avoid the biases and improve the accuracy of clustering. The difficulty in developing Ensemble Clustering methods is to combine external information (provided by input clusterings) with internal information (i.e. characteristics of given data) effectively to improve the accuracy of clustering. The work presented in this thesis focuses on enhancing the clustering accuracy of Ensemble Clustering by employing heuristic optimisation techniques to achieve a robust combination of relevant information during the consensus clustering stage. Two novel heuristic optimisation-based Ensemble Clustering methods, Multi-Optimisation Consensus Clustering (MOCC) and K-Ants Consensus Clustering (KACC), are developed and introduced in this thesis. These methods utilise two heuristic optimisation algorithms (Simulated Annealing and Ant Colony Optimisation) for their Ensemble Clustering frameworks, and have been proved to outperform other methods in the area. The extensive experimental results, together with a detailed analysis, will be presented in this thesis

Particle Swarm Optimization for the Clustering of Wireless Sensors

Clustering is necessary for data aggregation, hierarchical routing, optimizing sleep patterns, election of extremal sensors, optimizing coverage and resource allocation, reuse of frequency bands and codes, and conserving energy. Optimal clustering is typically an NP-hard problem. Solutions to NP-hard problems involve searches through vast spaces of possible solutions. Evolutionary algorithms have been applied successfully to a variety of NP-hard problems. We explore one such approach, Particle Swarm Optimization (PSO), an evolutionary programming technique where a \u27swarm\u27 of test solutions, analogous to a natural swarm of bees, ants or termites, is allowed to interact and cooperate to find the best solution to the given problem. We use the PSO approach to cluster sensors in a sensor network. The energy efficiency of our clustering in a data-aggregation type sensor network deployment is tested using a modified LEACH-C code. The PSO technique with a recursive bisection algorithm is tested against random search and simulated annealing; the PSO technique is shown to be robust. We further investigate developing a distributed version of the PSO algorithm for clustering optimally a wireless sensor network

Analyzing complex data using domain constraints

Data-driven research approaches are becoming increasingly popular in a growing number of scientific disciplines. While a data-driven research approach can yield superior results, generating the required data can be very costly. This frequently leads to small and complex data sets, in which it is impossible to rely on volume alone to compensate for all shortcomings of the data. To counter this problem, other reliable sources of information must be incorporated. In this work, domain knowledge, as a particularly reliable type of additional information, is used to inform data-driven analysis methods. This domain knowledge is represented as constraints on the possible solutions, which the presented methods can use to inform their analysis. It focusses on spatial constraints as a particularly common type of constraint, but the proposed techniques are general enough to be applied to other types of constraints. In this thesis, new methods using domain constraints for data-driven science applications are discussed. These methods have applications in feature evaluation, route database repair, and Gaussian Mixture modeling of spatial data. The first application focuses on feature evaluation. The presented method receives two representations of the same data: one as the intended target and the other for investigation. It calculates a score indicating how much the two representations agree. A presented application uses this technique to compare a reference attribute set with different subsets to determine the importance and relevance of individual attributes. A second technique analyzes route data for constraint compliance. The presented framework allows the user to specify constraints and possible actions to modify the data. The presented method then uses these inputs to generate a version of the data, which agrees with the constraints, while otherwise reducing the impact of the modifications as much as possible. Two extensions of this schema are presented: an extension to continuously valued costs, which are minimized, and an extension to constraints involving more than one moving object. Another addressed application area is modeling of multivariate measurement data, which was measured at spatially distributed locations. The spatial information recorded with the data can be used as the basis for constraints. This thesis presents multiple approaches to building a model of this kind of data while complying with spatial constraints. The first approach is an interactive tool, which allows domain scientists to generate a model of the data, which complies with their knowledge about the data. The second is a Monte Carlo approach, which generates a large number of possible models, tests them for compliance with the constraints, and returns the best one. The final two approaches are based on the EM algorithm and use different ways of incorporating the information into their models. At the end of the thesis, two applications of the models, which have been generated in the previous chapter, are presented. The first is prediction of the origin of samples and the other is the visual representation of the extracted models on a map. These tools can be used by domain scientists to augment their tried and tested tools. The developed techniques are applied to a real-world data set collected in the archaeobiological research project FOR 1670 (Transalpine mobility and cultural transfer) of the German Science Foundation. The data set contains isotope ratio measurements of samples, which were discovered at archaeological sites in the Alps region of central Europe. Using the presented data analysis methods, the data is analyzed to answer relevant domain questions. In a first application, the attributes of the measurements are analyzed for their relative importance and their ability to predict the spatial location of samples. Another presented application is the reconstruction of potential migration routes between the investigated sites. Then spatial models are built using the presented modeling approaches. Univariate outliers are determined and used to predict locations based on the generated models. These are cross-referenced with the recorded origins. Finally, maps of the isotope distribution in the investigated regions are presented. The described methods and demonstrated analyses show that domain knowledge can be used to formulate constraints that inform the data analysis process to yield valid models from relatively small data sets and support domain scientists in their analyses.Datengetriebene Forschungsansätze werden für eine wachsende Anzahl von wissenschaftlichen Disziplinen immer wichtiger. Obwohl ein datengetriebener Forschungsansatz bessere Ergebnisse erzielen kann, kann es sehr teuer sein die notwendigen Daten zu gewinnen. Dies hat häufig zur Folge, dass kleine und komplexe Datensätze entstehen, bei denen es nicht möglich ist sich auf die Menge der Datenpunkte zu verlassen um Probleme bei der Analyse auszugleichen. Um diesem Problem zu begegnen müssen andere Informationsquellen verwendet werden. Fachwissen als eine besonders zuverlässige Quelle solcher Informationen kann herangezogen werden, um die datengetriebenen Analysemethoden zu unterstützen. Dieses Fachwissen wird ausgedrückt als Constraints (Nebenbedingungen) der möglichen Lösungen, die die vorgestellten Methoden benutzen können um ihre Analyse zu steuern. Der Fokus liegt dabei auf räumlichen Constraints als eine besonders häufige Art von Constraints, aber die vorgeschlagenen Methoden sind allgemein genug um auf andere Arte von Constraints angewendet zu werden. Es werden neue Methoden diskutiert, die Fachwissen für datengetriebene wissenschaftliche Anwendungen verwenden. Diese Methoden haben Anwendungen auf Feature-Evaluation, die Reparatur von Bewegungsdatenbanken und auf Gaussian-Mixture-Modelle von räumlichen Daten. Die erste Anwendung betrifft Feature-Evaluation. Die vorgestellte Methode erhält zwei Repräsentationen der selben Daten: eine als Zielrepräsentation und eine zur Untersuchung. Sie berechnet einen Wert, der aussagt, wie einig sich die beiden Repräsentationen sind. Eine vorgestellte Anwendung benutzt diese Technik um eine Referenzmenge von Attributen mit verschiedenen Untermengen zu vergleichen, um die Wichtigkeit und Relevanz einzelner Attribute zu bestimmen. Eine zweite Technik analysiert die Einhaltung von Constraints in Bewegungsdaten. Das präsentierte Framework erlaubt dem Benutzer Constraints zu definieren und mögliche Aktionen zur Veränderung der Daten anzuwenden. Die präsentierte Methode benutzt diese Eingaben dann um eine neue Variante der Daten zu erstellen, die die Constraints erfüllt ohne die Datenbank mehr als notwendig zu verändern. Zwei Erweiterungen dieser Grundidee werden vorgestellt: eine Erweiterung auf stetige Kostenfunktionen, die minimiert werden, und eine Erweiterung auf Bedingungen, die mehr als ein bewegliches Objekt betreffen. Ein weiteres behandeltes Anwendungsgebiet ist die Modellierung von multivariaten Messungen, die an räumlich verteilten Orten gemessen wurden. Die räumliche Information, die zusammen mit diesen Daten erhoben wurde, kann als Grundlage genutzt werden um Constraints zu formulieren. Mehrere Ansätze zum Erstellen von Modellen auf dieser Art von Daten werden vorgestellt, die räumliche Constraints einhalten. Der erste dieser Ansätze ist ein interaktives Werkzeug, das Fachwissenschaftlern dabei hilft, Modelle der Daten zu erstellen, die mit ihrem Wissen über die Daten übereinstimmen. Der zweite ist eine Monte-Carlo-Simulation, die eine große Menge möglicher Modelle erstellt, testet ob sie mit den Constraints übereinstimmen und das beste Modell zurückgeben. Zwei letzte Ansätze basieren auf dem EM-Algorithmus und benutzen verschiedene Arten diese Information in das Modell zu integrieren. Am Ende werden zwei Anwendungen der gerade vorgestellten Modelle vorgestellt. Die erste ist die Vorhersage der Herkunft von Proben und die andere ist die grafische Darstellung der erstellten Modelle auf einer Karte. Diese Werkzeuge können von Fachwissenschaftlern benutzt werden um ihre bewährten Methoden zu unterstützen. Die entwickelten Methoden werden auf einen realen Datensatz angewendet, der von dem archäo-biologischen Forschungsprojekt FOR 1670 (Transalpine Mobilität und Kulturtransfer der Deutschen Forschungsgemeinschaft erhoben worden ist. Der Datensatz enthält Messungen von Isotopenverhältnissen von Proben, die in archäologischen Fundstellen in den zentraleuropäischen Alpen gefunden wurden. Die präsentierten Datenanalyse-Methoden werden verwendet um diese Daten zu analysieren und relevante Forschungsfragen zu klären. In einer ersten Anwendung werden die Attribute der Messungen analysiert um ihre relative Wichtigkeit und ihre Fähigkeit zu bewerten, die räumliche Herkunft der Proben vorherzusagen. Eine weitere vorgestellte Anwendung ist die Wiederherstellung von möglichen Migrationsrouten zwischen den untersuchten Fundstellen. Danach werden räumliche Modelle der Daten unter Verwendung der vorgestellten Methoden erstellt. Univariate Outlier werden bestimmt und ihre möglich Herkunft basierend auf der erstellten Karte wird bestimmt. Die vorhergesagte Herkunft wird mit der tatsächlichen Fundstelle verglichen. Zuletzt werden Karten der Isotopenverteilung der untersuchten Region vorgestellt. Die beschriebenen Methoden und vorgestellten Analysen zeigen, dass Fachwissen verwendet werden kann um Constraints zu formulieren, die den Datenanalyseprozess unterstützen, um gültige Modelle aus relativ kleinen Datensätzen zu erstellen und Fachwissenschaftler bei ihren Analysen zu unterstützen

Global Optimization strategies for two-mode clustering

Two-mode clustering is a relatively new form of clustering that clusters both rows and columns of a data matrix. To do so, a criterion similar to k-means is optimized. However, it is still unclear which optimization method should be used to perform two-mode clustering, as various methods may lead to non-global optima. This paper reviews and compares several optimization methods for two-mode clustering. Several known algorithms are discussed and a new, fuzzy algorithm is introduced. The meta-heuristics Multistart, Simulated Annealing, and Tabu Search are used in combination with these algorithms. The new, fuzzy algorithm is based on the fuzzy c-means algorithm of Bezdek (1981) and the Fuzzy Steps approach to avoid local minima of Heiser and Groenen (1997) and Groenen and Jajuga (2001). The performance of all methods is compared in a large simulation study. It is found that using a Multistart meta-heuristic in combination with a two-mode k-means algorithm or the fuzzy algorithm often gives the best results. Finally, an empirical data set is used to give a practical example of two-mode clustering.algorithms;fuzzy clustering;multistart;simulated annealing;simulation;tabu search;two-mode clustering

Median evidential c-means algorithm and its application to community detection

Median clustering is of great value for partitioning relational data. In this paper, a new prototype-based clustering method, called Median Evidential C-Means (MECM), which is an extension of median c-means and median fuzzy c-means on the theoretical framework of belief functions is proposed. The median variant relaxes the restriction of a metric space embedding for the objects but constrains the prototypes to be in the original data set. Due to these properties, MECM could be applied to graph clustering problems. A community detection scheme for social networks based on MECM is investigated and the obtained credal partitions of graphs, which are more refined than crisp and fuzzy ones, enable us to have a better understanding of the graph structures. An initial prototype-selection scheme based on evidential semi-centrality is presented to avoid local premature convergence and an evidential modularity function is defined to choose the optimal number of communities. Finally, experiments in synthetic and real data sets illustrate the performance of MECM and show its difference to other methods

Meta Clustering

Clustering is ill-defined. Unlike supervised learning where labels lead to crisp performance criteria such as accuracy and squared error, clustering quality depends on how the clusters will be used. Devising clustering criteria that capture what users need is difficult. Most clustering algorithms search for one optimal clustering based on a pre-specified clustering criterion. Once that clustering has been determined, no further clusterings are examined. Our approach differs in that we search for many alternate reasonable clusterings of the data, and then allow users to select the clustering(s) that best fit their needs. Any reasonable partitioning of the data is potentially useful for some purpose, regardless of whether or not it is optimal according to a specific clustering criterion. Our approach first finds a variety of reasonable clusterings. It then clusters this diverse set of clusterings so that users must only examine a small number of qualitatively different clusterings. In this paper, we present methods for automatically generating a diverse set of alternate clusterings, as well as methods for grouping clusterings into meta clusters. We evaluate meta clustering on four test problems, and then apply meta clustering to two case studies. Surprisingly, clusterings that would be of most interest to users often are not very compact clusterings