Symmetry in Domination for Hypergraphs with Choice

Berenhaut, Kenneth S.; Lidral-Porter, Brendan P.; Schoen, Theodore H.; Webb, Kyle P.

doi:10.3390/sym9030046

Cited by 1 publication

(7 citation statements)

References 21 publications

(31 reference statements)

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“…Mandal et al [35] consider hierarchical clustering and measure the goodness of dendrograms. Their goals are different to ours, which can be seen as a scalable redesign of [9].…”

Section: Presentations Of Ranking Systemsmentioning

confidence: 92%

“…Why do we favor 3-concordant ranking systems over concordant ones? For algorithms such as rank-based linkage (Algorithm 1) and partitioned local depth [9] which study triangles in which a particular 0-simplex (i.e. object pair) is a source, the extra generality is welcome, and carries no cost because cycles longer than three do not matter.…”

Section: Presentations Of Ranking Systemsmentioning

confidence: 99%

“…In this section we deal with practical graph computation on billion-scale edge sets. The number of voter triangles for a ranking system on n objects is O(n 3 ), which is impractical at this scale; indeed, cubic complexity limits the application of the related PaLD algorithm [9]. To dodge this burden, we work with a K-nearest-neighbor approximation 1 to the ranking system.…”

Section: Hierarchical Clustering Based On Linkagementioning

confidence: 99%

“…In-sway. The notion of in-sway is intended to convey relative closeness of two objects in S, with respect to a ranking system, just as cohesion does for partitioned local depth [9]. Computation of in-sway is the driving principle of rank-based linkage: see Algorithm 1.…”

Section: 2mentioning

confidence: 99%

“…Partitioned local depth versus rank-based linkage. Rank-based linkage was inspired partly by partitioned local depth (PaLD) [9]. Figure 6 illustrates the cluster graph produced by PaLD, run against the same data from Table 1 as was used to illustrate rankbased linkage in Figure 3.…”

Section: 4mentioning

confidence: 99%

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Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Darling¹,

Grilliette²,

Logan³

2023

Preprint

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Rank-based linkage is a new tool for summarizing a collection S of objects according to their relationships. These objects are not mapped to vectors, and "similarity" between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on S. Rankbased linkage is applied to the K-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected K-nearest neighbor graph on S. In |S|K 2 steps it builds an edge-weighted linkage graph (S, L, σ) where σ({x, y}) is called the in-sway between objects x and y. Take L t to be the links whose in-sway is at least t, and partition S into components of the graph (S, L t ), for varying t. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not "rip apart" the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.

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“…Mandal et al [35] consider hierarchical clustering and measure the goodness of dendrograms. Their goals are different to ours, which can be seen as a scalable redesign of [9].…”

Section: Presentations Of Ranking Systemsmentioning

confidence: 92%

Section: Presentations Of Ranking Systemsmentioning

confidence: 99%

Section: Hierarchical Clustering Based On Linkagementioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

See 3 more Smart Citations