Hierarchical Quasi-Clustering Methods for Asymmetric Networks

Pattern Recognition, 1980

The present simulation study examined the ability of four hierarchical clustering algorithms to recover the true structure in data sets which satisfied both the ultrametric inequality and the structural model of the clustering procedures. The results indicated that the rank order performance of the four methods differed markedly from the rank order generally found in multivariate normal mixture model studies. The differing rank orders demonstrates a lack of robustness of the algorithms over alternative conceptualizations of cluster structure.

Advances in Mathematics

A hierarchical clustering method is stable if small perturbations on the data set produce small perturbations in the result. This perturbations are measured using the Gromov-Hausdorff metric. We study the problem of stability on linkage-based hierarchical clustering methods. We obtain that, under some basic conditions, standard linkage-based methods are semi-stable. This means that they are stable if the input data is close enough to an ultrametric space. We prove that, apart from exotic examples, introducing any unchaining condition in the algorithm always produces unstable methods.

2009

In the paper an adaptation of the hierarchical clustering with relational constraints approach proposed by Ferligoj and Batagelj (1982,1983) to large data sets is presented. To obtain an efficient algorithm for large networks, we: (1) compute the dissimilarities between units (vertices of network) only for endpoints of existing links of constraining relation; (2) define the dissimilarities between clusters based only on the dissimilarities of the corresponding links and derive the update relations. We also show that for selected dissimilarities between clusters, the Bruynooghe reducibility property holds. This allows us to speed-up the hierarchical clustering procedure by using the RNN (reciprocal nearest neighbors) approach. The developed algorithms are implemented in Pajek – a program for analysis and visualization of large networks. The proposed approach is illustrated by clustering of US counties.

Journal of Physics: Conference Series

The increasing needs of clustering massive datasets and the high cost of running clustering algorithms poses difficult problems for users. In this context it is important to determine if a data set is clusterable, that is, it may be partitioned efficiently into welldifferentiated groups containing similar objects. We approach data clusterability from an ultrametric-based perspective. A novel approach to determine the ultrametricity of a dataset is proposed via a special type of matrix product, which allows us to evaluate the clusterability of the dataset. Furthermore, we show that by applying our technique to a dissimilarity space will generate the subdominant ultrametric of the dissimilarity.

Journal of Classification, 2016

In the election of a hierarchical clustering method, theoretic properties may give some insight to determine which method is the most suitable to treat a clustering problem. Herein, we study some basic properties of two hierarchical clustering methods: α-unchaining single linkage or SL(α) and a modified version of this one, SL * (α). We compare the results with the properties satisfied by the classical linkage-based hierarchical clustering methods.

Advanced studies in theoretical and applied econometrics, 2018

In this chapter, first a certain number of rigorous definitions are given, this being all the more necessary as the literature abounds in contradictory definitions and sloppy use of terms; for more details about this contention, one is referred to Paelinck (2004). The second part of this chapter describes some relevant clustering methods, including a number of recent analytical developments. For links with transport problems, again consult Paelinck (2004). The nonstandard treatment in this chapter contributes selected formal additions to the analysis literature. 16.1 An Axiomatic Basis Axioms are statements assumed to be true that form the foundation for deriving theorems about the context to which they apply. The context of interest here concerns spatial economic landscapes. 16.1.1 Clusters Assume a set S of arbitrary objects or elements; this set is structured by a distance relation (for a definition, see Paelinck and Vossen 1983, pp. 39-52), such that it is a metric space. A subset of S, S* may be called a cluster if 8a 2 S * ∃b 2 S * j d a; b ð Þ d * , ð16:1Þ where d is any distance measure. In other words, expression (16.1) means that in order to identify a cluster, one needs to select an S* from S such that every point in it

IEEE Transactions on Signal and Information Processing over Networks, 2017

A novel method to obtain hierarchical and overlapping clusters from network data-i.e., a set of nodes endowed with pairwise dissimilarities-is presented. The introduced method is hierarchical in the sense that it outputs a nested collection of groupings of the node set depending on the resolution or degree of similarity desired, and it is overlapping since it allows nodes to belong to more than one group. Our construction is rooted on the facts that a hierarchical (non-overlapping) clustering of a network can be equivalently represented by a finite ultrametric space and that a convex combination of ultrametrics results in a cut metric. By applying a hierarchical (non-overlapping) clustering method to multiple dithered versions of a given network and then convexly combining the resulting ultrametrics, we obtain a cut metric associated to the network of interest. We then show how to extract a hierarchical overlapping clustering structure from the aforementioned cut metric. Furthermore, the so-called overlapping function is presented as a tool for gaining insights about the data by identifying meaningful resolutions of the obtained hierarchical structure. Additionally, we explore hierarchical overlapping quasi-clustering methods that preserve the asymmetry of the data contained in directed networks. Finally, the presented method is illustrated via synthetic and real-world classification problems including handwritten digit classification and authorship attribution of famous plays.

ArXiv, 2017

While there has been much interest in adapting conventional clustering procedures---and in higher dimensions, persistent homology methods---to directed networks, little is known about the convergence of such methods. In order to even formulate the problem of convergence for such methods, one needs to stipulate a reasonable model for a directed network together with a flexible sampling theory for such a model. In this paper we propose and study a particular model of directed networks, and use this model to study the convergence of certain hierarchical clustering and persistent homology methods that accept any matrix of (possibly asymmetric) pairwise relations as input and produce dendrograms and persistence barcodes as outputs. We show that as points are sampled from some probability distribution, the output of each method converges almost surely to a dendrogram/barcode depending on the structure of the distribution.

Environment and Planning A, 1982

Connections between hierarchical clustering and the sedation of objects along a continuum that depend on the patterning of entries in a proximity matrix are pointed out. Based on the similarity between the central notion of an ultrametric in hierarchical clustering and what is called an anti-Robinson property in sedation, it is suggested that the two data-analysis procedures are compatible. Indeed, preliminary seriation of a proximity matrix may help verify the adequacy of the results obtained from a hierarchical clustering or suggest alternatives that may be better. A numerical example using data from the area of criminal justice completes the paper.