1,016 research outputs found
Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks
This paper considers networks where relationships between nodes are
represented by directed dissimilarities. The goal is to study methods for the
determination of hierarchical clusters, i.e., a family of nested partitions
indexed by a connectivity parameter, induced by the given dissimilarity
structures. Our construction of hierarchical clustering methods is based on
defining admissible methods to be those methods that abide by the axioms of
value - nodes in a network with two nodes are clustered together at the maximum
of the two dissimilarities between them - and transformation - when
dissimilarities are reduced, the network may become more clustered but not
less. Several admissible methods are constructed and two particular methods,
termed reciprocal and nonreciprocal clustering, are shown to provide upper and
lower bounds in the space of admissible methods. Alternative clustering
methodologies and axioms are further considered. Allowing the outcome of
hierarchical clustering to be asymmetric, so that it matches the asymmetry of
the original data, leads to the inception of quasi-clustering methods. The
existence of a unique quasi-clustering method is shown. Allowing clustering in
a two-node network to proceed at the minimum of the two dissimilarities
generates an alternative axiomatic construction. There is a unique clustering
method in this case too. The paper also develops algorithms for the computation
of hierarchical clusters using matrix powers on a min-max dioid algebra and
studies the stability of the methods proposed. We proved that most of the
methods introduced in this paper are such that similar networks yield similar
hierarchical clustering results. Algorithms are exemplified through their
application to networks describing internal migration within states of the
United States (U.S.) and the interrelation between sectors of the U.S. economy.Comment: This is a largely extended version of the previous conference
submission under the same title. The current version contains the material in
the previous version (published in ICASSP 2013) as well as material presented
at the Asilomar Conference on Signal, Systems, and Computers 2013, GlobalSIP
2013, and ICML 2014. Also, unpublished material is included in the current
versio
Common Visual Pattern Recognition Using Hierarchical Clustering Methods with Asymmetric Networks
In this paper we propose a new method for common visual pattern identi cation via Directed Graphs. For this we match common feature points between two images and then apply hierarchical clustering methods to one of them to discriminate between di erent visual patterns.
In order to achieve this last task we introduce a technique to obtain an asymmetric dissimilarity function AX(x; x1) between the nodes X of the network N = (X;Ax). For each node, the method weighs the distance between each node and the distance with all the other neighbours. A dendrogram is later obtained as the output of the hierarchical clustering method. Finally we show a criteria to select one of the multiple partitions that conform the dendrogram.Sociedad Argentina de Informática e Investigación Operativa (SADIO
Metric Representations Of Networks
The goal of this thesis is to analyze networks by first projecting them onto structured metric-like spaces -- governed by a generalized triangle inequality -- and then leveraging this structure to facilitate the analysis. Networks encode relationships between pairs of nodes, however, the relationship between two nodes can be independent of the other ones and need not be defined for every pair. This is not true for metric spaces, where the triangle inequality imposes conditions that must be satisfied by triads of distances and these must be defined for every pair of nodes. In general terms, this additional structure facilitates the analysis and algorithm design in metric spaces. In deriving metric projections for networks, an axiomatic approach is pursued where we encode as axioms intuitively desirable properties and then seek for admissible projections satisfying these axioms. Although small variations are introduced throughout the thesis, the axioms of projection -- a network that already has the desired metric structure must remain unchanged -- and transformation -- when reducing dissimilarities in a network the projected distances cannot increase -- shape all of the axiomatic constructions considered. Notwithstanding their apparent weakness, the aforementioned axioms serve as a solid foundation for the theory of metric representations of networks.
We begin by focusing on hierarchical clustering of asymmetric networks, which can be framed as a network projection problem onto ultrametric spaces. We show that the set of admissible methods is infinite but bounded in a well-defined sense and state additional desirable properties to further winnow the admissibility landscape. Algorithms for the clustering methods developed are also derived and implemented. We then shift focus to projections onto generalized q-metric spaces, a parametric family containing among others the (regular) metric and ultrametric spaces. A uniqueness result is shown for the projection of symmetric networks whereas for asymmetric networks we prove that all admissible projections are contained between two extreme methods. Furthermore, projections are illustrated via their implementation for efficient search and data visualization. Lastly, our analysis is extended to encompass projections of dioid spaces, natural algebraic generalizations of weighted networks
Common Visual Pattern Recognition Using Hierarchical Clustering Methods with Asymmetric Networks
In this paper we propose a new method for common visual pattern identi cation via Directed Graphs. For this we match common feature points between two images and then apply hierarchical clustering methods to one of them to discriminate between di erent visual patterns.
In order to achieve this last task we introduce a technique to obtain an asymmetric dissimilarity function AX(x; x1) between the nodes X of the network N = (X;Ax). For each node, the method weighs the distance between each node and the distance with all the other neighbours. A dendrogram is later obtained as the output of the hierarchical clustering method. Finally we show a criteria to select one of the multiple partitions that conform the dendrogram.Sociedad Argentina de Informática e Investigación Operativa (SADIO
Incompatibility boundaries for properties of community partitions
We prove the incompatibility of certain desirable properties of community
partition quality functions. Our results generalize the impossibility result of
[Kleinberg 2003] by considering sets of weaker properties. In particular, we
use an alternative notion to solve the central issue of the consistency
property. (The latter means that modifying the graph in a way consistent with a
partition should not have counterintuitive effects). Our results clearly show
that community partition methods should not be expected to perfectly satisfy
all ideally desired properties.
We then proceed to show that this incompatibility no longer holds when
slightly relaxed versions of the properties are considered, and we provide in
fact examples of simple quality functions satisfying these relaxed properties.
An experimental study of these quality functions shows a behavior comparable to
established methods in some situations, but more debatable results in others.
This suggests that defining a notion of good partition in communities probably
requires imposing additional properties.Comment: 17 pages, 3 figure
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