Arc Diagrams Research Articles

Constraint-Based Breakpoints for Responsive Visualization Design and Development.

This paper introduces constraint-based breakpoints, a technique for designing responsive visualizations for a wide variety of screen sizes and datasets. Breakpoints in responsive visualization define when different visualization designs are shown. Conventionally, breakpoints are static, pre-defined widths, and as such do not account for changes to the visualized dataset or visualization parameters. To guarantee readability and efficient use of space across datasets, these static breakpoints would require manual updates. Constraint-based breakpoints solve this by evaluating visualization-specific constraints on the size of visual elements, overlapping elements, and the aspect ratio of the visualization and available space. Once configured, a responsive visualization with constraint-based breakpoints can adapt to different screen sizes for any dataset. We describe a framework that guides designers in creating a stack of visualization designs for different display sizes and defining constraints for each of these designs. We demonstrate constraint-based breakpoints for different data types and their visualizations: geographic data (choropleth map, proportional circle map, Dorling cartogram, hexagonal grid map, bar chart, waffle chart), network data (node-link diagram, adjacency matrix, arc diagram), and multivariate data (scatterplot, heatmap). Interactive demos and supplemental material are available at responsive-vis.github.io/breakpoints.

Improved Visual Saliency of Graph Clusters with Orderable Node-Link Layouts.

Graphs are often used to model relationships between entities. The identification and visualization of clusters in graphs enable insight discovery in many application areas, such as life sciences and social sciences. Force-directed graph layouts promote the visual saliency of clusters, as they bring adjacent nodes closer together, and push non-adjacent nodes apart. At the same time, matrices can effectively show clusters when a suitable row/column ordering is applied, but are less appealing to untrained users not providing an intuitive node-link metaphor. It is thus worth exploring layouts combining the strengths of the node-link metaphor and node ordering. In this work, we study the impact of node ordering on the visual saliency of clusters in orderable node-link diagrams, namely radial diagrams, arc diagrams and symmetric arc diagrams. Through a crowdsourced controlled experiment, we show that users can count clusters consistently more accurately, and to a large extent faster, with orderable node-link diagrams than with three state-of-the art force-directed layout algorithms, i.e., 'Linlog', 'Backbone' and 'sfdp'. The measured advantage is greater in case of low cluster separability and/or low compactness.

E-RNA: a collection of web-servers for the prediction and visualisation of RNA secondary structure and their functional features.

e-RNA is a collection of web-servers for the prediction and visualisation of RNA secondary structures and their functional features, including in particular RNA-RNA interactions. In this updated version, we have added novel tools for RNA secondary structure prediction and have significantly updated the visualisation functionality. The new method CoBold can identify transient RNA structure features and their potential functional effects on a known RNA structure during co-transcriptional structure formation. New tool ShapeSorter can predict evolutionarily conserved RNA secondary structure features while simultaneously taking experimental SHAPE probing evidence into account. The web-server R-Chie which visualises RNA secondary structure information in terms of arc diagrams, can now be used to also visualise and intuitively compare RNA-RNA, RNA-DNA and DNA-DNA interactions alongside multiple sequence alignments and quantitative information. The prediction generated by any method in e-RNA can be readily visualised on the web-server. For completed tasks, users can download their results and readily visualise them later on with R-Chie without having to re-run the predictions. e-RNA can be found at http://www.e-rna.org.

Enumeration of symmetric arc diagrams

We give recurrence relations for the enumeration of symmetric elements within four classes of arc diagrams corresponding to certain involutions and set partitions whose blocks contain no consecutive integers. These arc diagrams are motivated by the study of RNA secondary structures. For example, classic RNA secondary structures correspond to 3412-avoiding involutions with no adjacent transpositions, and structures with base triples may be represented as partitions with crossings. Our results rely on combinatorial arguments. In particular, we use Motzkin paths to describe noncrossing arc diagrams that have no arc connecting two adjacent nodes, and we give an explicit bijection to ternary words whose length coincides with the sum of their digits. We also discuss the asymptotic behavior of some of the sequences considered here in order to quantify the extremely low probability of finding symmetric structures with a large number of nodes.

RNA foldings, oriented stuck knots, and state sum invariants

We extend the quandle cocycle invariant to the context of stuck links. More precisely, we define an invariant of stuck links by assigning Boltzmann weights at both classical and stuck crossings. As an application, we define a single-variable and a two-variable polynomial invariant of stuck links. Furthermore, we define a single-variable and two-variable polynomial invariant of arc diagrams of RNA foldings. We provide explicit computations of the new invariants.

The Canonical Complex of the Weak Order

We define and study the canonical complex of a finite semidistributive lattice L. It is the simplicial complex on the join or meet irreducible elements of L which encodes each interval of L by recording the canonical join representation of its bottom element and the canonical meet representation of its top element. This complex behaves properly with respect to lattice quotients of L, in the sense that the canonical complex of a quotient of L is the subcomplex of the canonical complex of L induced by the join or meet irreducibles of L uncontracted in the quotient. We then describe combinatorially the canonical complex of the weak order on permutations in terms of semi-crossing arc bidiagrams, formed by the superimposition of two non-crossing arc diagrams of N. Reading. We provide explicit direct bijections between the semi-crossing arc bidiagrams and the weak order interval posets of G. Châtel, V. Pilaud and V. Pons. Finally, we provide an algorithm to describe the Kreweras maps in any lattice quotient of the weak order in terms of semi-crossing arc bidiagrams.

Diagrammatic Modelling Tools for Grounded Theory Research: The Implementation of a Multi-Representational Approach

Grounded Theory (GT) researchers have an ever-expanding palette of digital tools available to further analyse complex phenomena with interrelated data sets. However, few GT researchers have systematically examined how the use of diagramming tools can enhance analysis. To advance the analytical process of GT, this study develops a multi-representational approach that integrates with research design. After diagrams supportive of GT are identified for their potential improvements to the analytical process, the research focuses on the experience of employing three diagramming tools (Flourish, Observable, and Pajek) in developing two complementary diagrams (Network and Arc diagrams). The use of these tools for analysis is explained in detail for conducting extensive constructivist GT study; illustrated via a case study examining a century of innovation in hospital design. Via this multiple-source case study, this paper demonstrates how the sagacious deployment of diagramming tools, when carefully aligned to research objectives, can complement GT analysis by facilitating systematic thinking and holistic interpretation of hidden patterns.

Arc diagrams and 2-term simple-minded collections of preprojective algebras of type A

We study an explicit description of semibricks and 2-term simple-minded collections over preprojective algebras of type A via arc diagrams. We provide a bijection between the set of noncrossing arc diagrams (resp. the set of double arc diagrams), which is in bijective correspondence with elements of the symmetric group, and the set of semibricks (resp. the set of 2-term simple-minded collections) over the algebra. Moreover we define a mutation and a partial order on the set of double arc diagrams. In particular, we obtain a poset isomorphism between the symmetric group and the set of 2-term simple-minded collections. As an application of our results, we study semibricks of some quotient algebras of the preprojective algebras of type A and we reprove some important results shown by the other authors.

ONQUADRO: a database of experimentally determined quadruplex structures.

ONQUADRO is an advanced database system that supports the study of the structures of canonical and non-canonical quadruplexes. It combines a relational database that collects comprehensive information on tetrads, quadruplexes, and G4-helices; programs to compute structure parameters and visualise the data; scripts for statistical analysis; automatic updates and newsletter modules; and a web application that provides a user interface. The database is a self-updating resource, with new information arriving once a week. The preliminary data are downloaded from the Protein Data Bank, processed, annotated, and completed. As of August 2021, ONQUADRO contains 1,661 tetrads, 518 quadruplexes, and 30 G4-helices found in 467 experimentally determined 3D structures of nucleic acids. Users can view and download their description: sequence, secondary structure (dot-bracket, classical diagram, arc diagram), tertiary structure (ball-and-stick, surface or vdw-ball model, layer diagram), planarity, twist, rise, chi angle (value and type), loop characteristics, strand directionality, metal ions, ONZ, and Webba da Silva classification (the latter by loop topology and tetrad combination), origin structure ID, assembly ID, experimental method, and molecule type. The database is freely available at https://onquadro.cs.put.poznan.pl/. It can be used on both desktop computers and mobile devices.

Counting Non-Crossing Permutations on Surfaces of any Genus

Given a surface with boundary and some points on the boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on the surface. If only bigons are allowed, then one obtains the notion of arc diagrams, whose enumeration is known to have a rich structure. We show that the count of polygon diagrams on surfaces with any genus and number of boundary components exhibits similar structure. In particular it is almost polynomial in the number of points on the boundary components, and the leading coefficients of those polynomials are intersection numbers on compactified moduli spaces of curves.

Semisimplicity of the DS functor for the orthosymplectic Lie superalgebra

We prove that the Duflo-Serganova functor DSx attached to an odd nilpotent element x of osp(m|2n) is semisimple, i.e. sends a semisimple representation M of osp(m|2n) to a semisimple representation of osp(m−2k|2n−2k) where k is the rank of x. We prove a closed formula for DSx(L(λ)) in terms of the arc diagram attached to λ.

Combinatorial generation via permutation languages. II. Lattice congruences

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.

Structural relation matching: an algorithm to identify structural patterns into RNAs and their interactions.

RNA molecules play crucial roles in various biological processes. Their three-dimensional configurations determine the functions and, in turn, influences the interaction with other molecules. RNAs and their interaction structures, the so-called RNA–RNA interactions, can be abstracted in terms of secondary structures, i.e., a list of the nucleotide bases paired by hydrogen bonding within its nucleotide sequence. Each secondary structure, in turn, can be abstracted into cores and shadows. Both are determined by collapsing nucleotides and arcs properly. We formalize all of these abstractions as arc diagrams, whose arcs determine loops. A secondary structure, represented by an arc diagram, is pseudoknot-free if its arc diagram does not present any crossing among arcs otherwise, it is said pseudoknotted. In this study, we face the problem of identifying a given structural pattern into secondary structures or the associated cores or shadow of both RNAs and RNA–RNA interactions, characterized by arbitrary pseudoknots. These abstractions are mapped into a matrix, whose elements represent the relations among loops. Therefore, we face the problem of taking advantage of matrices and submatrices. The algorithms, implemented in Python, work in polynomial time. We test our approach on a set of 16S ribosomal RNAs with inhibitors of Thermus thermophilus, and we quantify the structural effect of the inhibitors.

Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group

Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, these canonical join representations can be modeled in terms of arc diagrams. Moreover, these arc diagrams also serve as a model to understand quotient lattices of the weak order. A particularly well-behaved quotient lattice of the weak order is the well-known Tamari lattice, which appears in many seemingly unrelated areas of mathematics. The arc diagrams representing the members of the Tamari lattices are better known as noncrossing partitions. Recently, the Tamari lattices were generalized to parabolic quotients of the symmetric group. In this article, we undertake a structural investigation of these parabolic Tamari lattices, and explain how modified arc diagrams aid the understanding of these lattices.

RNAStructViz: graphical base pairing analysis.

We present a new graphical tool for RNA secondary structure analysis. The central feature is the ability to visually compare/contrast up to three base pairing configurations for a given sequence in a compact, standardized circular arc diagram layout. This is complemented by a built-in CT-style file viewer and radial layout substructure viewer which are directly linked to the arc diagram window via the zoom selection tool. Additional functionality includes the computation of some numerical information, and the ability to export images and data for later use. This tool should be of use to researchers seeking to better understand similarities and differences between structural alternatives for an RNA sequence. https://github.com/gtDMMB/RNAStructViz/wiki.

Гафниан некоторых трехпараметрических теплицевых матриц и совершенные паросочетания дуговых и хордовых диаграмм

We obtain an explicit formula for exact calculatingthe hafnian of 3-parameter Toeplitz matrices of aspecial type in polynomial time. We also give anasymptotic estimate for the hafnian of this type ofmatrices. Separately, we consider a case of non-negativeinteger parameters, when calculating the hafnianis equivalent to enumerating perfect matchings of arc and chord diagrams.

ARC: Angle-Retaining Chromaticity diagram for color constancy error analysis.

Color constancy algorithms are typically evaluated with a statistical analysis of the recovery angular error and the reproduction angular error between the estimated and ground truth illuminants. Such analysis provides information about only the magnitude of the errors, and not about their chromatic properties. We propose an Angle-Retaining Chromaticity diagram (ARC) for the visual analysis of the estimated illuminants and the corresponding errors. We provide both quantitative and qualitative proof of the superiority of ARC in preserving angular distances compared to other chromaticity diagrams, making it possible to quantify the reproduction and recovery errors in terms of Euclidean distances on a plane. We present two case studies for the application of the ARC diagram in the visualization of the ground truth illuminants of color constancy datasets, and the visual analysis of error distributions of color constancy algorithms.

Knotting probability of an arc diagram

The knotting probability of an arc diagram is defined as the quadruplet of four kinds of finner knotting probabilities which are invariant under a reasonable deformation containing an isomorphism on an arc diagram. In a separated paper, it is shown that every oriented spatial arc admits four kinds of unique arc diagrams up to isomorphisms determined from the spatial arc and the projection, so that the knotting probability of a spatial arc is defined. The definition of the knotting probability of an arc diagram uses the fact that every arc diagram induces a unique chord diagram representing a ribbon 2-knot. Then the knotting probability of an arc diagram is set to measure how many nontrivial ribbon genus 2 surface-knots occur from the chord diagram induced from the arc diagram. The conditions for an arc diagram with the knotting probability 0 and for an arc diagram with the knotting probability 1 are given together with some other properties and some examples.

MutaRNA: analysis and visualization of mutation-induced changes in RNA structure.

RNA molecules fold into complex structures as a result of intramolecular interactions between their nucleotides. The function of many non-coding RNAs and some cis-regulatory elements of messenger RNAs highly depends on their fold. Single-nucleotide variants (SNVs) and other types of mutations can disrupt the native function of an RNA element by altering its base pairing pattern. Identifying the effect of a mutation on an RNA’s structure is, therefore, a crucial step in evaluating the impact of mutations on the post-transcriptional regulation and function of RNAs within the cell. Even though a single nucleotide variation can have striking impacts on the structure formation, interpreting and comparing the impact usually needs expertise and meticulous efforts. Here, we present MutaRNA, a web server for visualization and interpretation of mutation-induced changes on the RNA structure in an intuitive and integrative fashion. To this end, probabilities of base pairing and position-wise unpaired probabilities of wildtype and mutated RNA sequences are computed and compared. Differential heatmap-like dot plot representations in combination with circular plots and arc diagrams help to identify local structure abberations, which are otherwise hidden in standard outputs. Eventually, MutaRNA provides a comprehensive and comparative overview of the mutation-induced changes in base pairing potentials and accessibility. The MutaRNA web server is freely available at http://rna.informatik.uni-freiburg.de/MutaRNA.

The canonical join complex of the Tamari lattice

In this paper, we study a simplicial complex on the elements of the Tamari lattice in types A and B called the canonical join complex. The canonical join representation of an element w in a lattice L is the unique lowest expression ⋁A for w, when such an expression exists. We say that the set A is a canonical join representation. The collection of all such subsets has the structure of an abstract simplicial complex called the canonical join complex of L. We realize the canonical join complex of the Tamari lattice as a complex of noncrossing arc diagrams, give a shelling order on its facets, and show that it is homotopy equivalent to a wedge of Catalan-many spheres.