Metrics in the Space of High Order Networks

Alejandro Ribeiro; Weiyu Huang

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Metrics in the Space of High Order Networks

Weiyu Huang, Alejandro Ribeiro

Social and Information Networks · Submitted 8 Jul 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that they are valid metrics in the spaces of high order dissimilarity and proximity networks modulo permutation isomorphisms.

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,975 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.10 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.15 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

This paper presents methods to compare high order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples. They can be considered as generalizations of conventional networks where only relationship functions between pairs are defined. Important properties between relationships of tuples of different lengths are established, particularly when relationships encode dissimilarities or proximities between nodes. Two families of distances are then introduced in the space of high order networks. The distances measure differences between networks. We prove that they are valid metrics in the spaces of high order dissimilarity and proximity networks modulo permutation isomorphisms. Practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers. The metrics succeed in identifying their respective collaboration patterns.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This paper presents methods to compare high order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples.	contribution 0.91	0.85	0.06	asserted 1.00
s1	They can be considered as generalizations of conventional networks where only relationship functions between pairs are defined.	background 1.00	0.10	0.03	asserted 0.55
s2	Important properties between relationships of tuples of different lengths are established, particularly when relationships encode dissimilarities or proximities between nodes.	contribution 0.50	0.71	0.06	asserted 1.00
s3	Two families of distances are then introduced in the space of high order networks.	contribution 0.91	0.91	0.06	asserted 1.00
s4	The distances measure differences between networks.	contribution 0.64	0.76	0.07	asserted 1.00
s5	We prove that they are valid metrics in the spaces of high order dissimilarity and proximity networks modulo permutation isomorphisms.	result 0.84	0.95	0.66	asserted 1.00
s6	Practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers.	method 0.68	0.88	0.05	asserted 0.87
s7	The metrics succeed in identifying their respective collaboration patterns.	result 0.95	0.89	0.27	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.85 · evidence from proof or derivation 0.97.

What was examined, and what was not

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arXiv listing abstract v2

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full text
figures and tables
references and citing sentences
relations to other papers

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P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

This paper presents methods to compare high order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples. They can be considered as generalizations of conventional networks where only relationship functions between pairs are defined. Important properties between relationships of tuples of different lengths are established, particularly when relationships encode dissimilarities or proximities between nodes. Two families of distances are then introduced in the space of high order networks. The distances measure differences between networks. We prove that they are valid metrics in the spaces of high order dissimilarity and proximity networks modulo permutation isomorphisms. Practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers. The metrics succeed in identifying their respective collaboration patterns.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Builds on 1 paper

These distances are built as generalizations of the pairwise distances in [ref] , which are themselves generalizations of the Gromov-Hausdorff distance between metric spaces [ref] .

Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks · Introduction · 0.98

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