Dynamic Euler Diagram Drawing

In this paper we describe a method to lay out a graph enhanced Euler diagram so that it looks similar to a previously drawn graph enhanced Euler diagram. This task is non-trivial when the underlying structures of the diagrams differ. In particular, if a structural change is made to an existing drawn diagram, our work enables the presentation of the new diagram with minor disruption to the user's mental map. As the new diagram can be generated from an abstract representation, its initial embedding may be very different from that of the original. We have developed comparison measures for Euler diagrams, integrated into a multicriteria optimizer, and applied a force model for associated graphs that attempts to move nodes towards their positions in the original layout. To further enhance the usability of the system, the transition between diagrams can be animated

Evaluating the Comprehension of Euler Diagrams

We describe an empirical investigation into layout criteria that can help with the comprehension of Euler diagrams. Euler diagrams are used to represent set inclusion in applications such as teaching set theory, database querying, software engineering, filing system organisation and bio-informatics. Research in automatically laying out Euler diagrams for use with these applications is at an early stage, and our work attempts to aid this research by informing layout designers about the importance of various Euler diagram aesthetic criteria. The three criteria under investigation were: contour jaggedness, zone area inequality and edge closeness. Subjects were asked to interpret diagrams with different combinations of levels for each of the criteria. Results for this investigation indicate that, within the parameters of the study, all three criteria are important for understanding Euler diagrams and we have a preliminary indication of the ordering of their importance

eulerForce: Force-directed Layout for Euler Diagrams

Euler diagrams use closed curves to represent sets and their relationships. They facilitate set analysis, as humans tend to perceive distinct regions when closed curves are drawn on a plane. However, current automatic methods often produce diagrams with irregular, non-smooth curves that are not easily distinguishable. Other methods restrict the shape of the curve to for instance a circle, but such methods cannot draw an Euler diagram with exactly the required curve intersections for any set relations. In this paper, we present eulerForce, as the first method to adopt a force-directed approach to improve the layout and the curves of Euler diagrams generated by current methods. The layouts are improved in quick time. Our evaluation of eulerForce indicates the benefits of a force-directed approach to generate comprehensible Euler diagrams for any set relations in relatively fast time

The State-of-the-Art of Set Visualization

Sets comprise a generic data model that has been used in a variety of data analysis problems. Such problems involve analysing and visualizing set relations between multiple sets defined over the same collection of elements. However, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into six main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address these challenges. Further resources on set visualization are available at http://www.setviz.net

MetroSets: Visualizing Sets as Metro Maps

We propose MetroSets, a new, flexible online tool for visualizing set systems using the metro map metaphor. We model a given set system as a hypergraph $\mathcal{H} = (V, \mathcal{S})$, consisting of a set $V$ of vertices and a set $\mathcal{S}$, which contains subsets of $V$ called hyperedges. Our system then computes a metro map representation of $\mathcal{H}$, where each hyperedge $E$ in $\mathcal{S}$ corresponds to a metro line and each vertex corresponds to a metro station. Vertices that appear in two or more hyperedges are drawn as interchanges in the metro map, connecting the different sets. MetroSets is based on a modular 4-step pipeline which constructs and optimizes a path-based hypergraph support, which is then drawn and schematized using metro map layout algorithms. We propose and implement multiple algorithms for each step of the MetroSet pipeline and provide a functional prototype with \new{easy-to-use preset configurations.} % many real-world datasets. Furthermore, \new{using several real-world datasets}, we perform an extensive quantitative evaluation of the impact of different pipeline stages on desirable properties of the generated maps, such as octolinearity, monotonicity, and edge uniformity.Comment: 19 pages; accepted for IEEE INFOVIS 2020; for associated live system, see http://metrosets.ac.tuwien.ac.a

Visualizing Set Relations and Cardinalities Using Venn and Euler Diagrams

In medicine, genetics, criminology and various other areas, Venn and Euler diagrams are used to visualize data set relations and their cardinalities. The data sets are represented by closed curves and the data set relationships are depicted by the overlaps between these curves. Both the sets and their intersections are easily visible as the closed curves are preattentively processed and form common regions that have a strong perceptual grouping effect. Besides set relations such as intersection, containment and disjointness, the cardinality of the sets and their intersections can also be depicted in the same diagram (referred to as area-proportional) through the size of the curves and their overlaps. Size is a preattentive feature and so similarities, differences and trends are easily identified. Thus, such diagrams facilitate data analysis and reasoning about the sets. However, drawing these diagrams manually is difficult, often impossible, and current automatic drawing methods do not always produce appropriate diagrams. This dissertation presents novel automatic drawing methods for different types of Euler diagrams and a user study of how such diagrams can help probabilistic judgement. The main drawing algorithms are: eulerForce, which uses a force-directed approach to lay out Euler diagrams; eulerAPE, which draws area-proportional Venn diagrams with ellipses. The user study evaluated the effectiveness of area- proportional Euler diagrams, glyph representations, Euler diagrams with glyphs and text+visualization formats for Bayesian reasoning, and a method eulerGlyphs was devised to automatically and accurately draw the assessed visualizations for any Bayesian problem. Additionally, analytic algorithms that instantaneously compute the overlapping areas of three general intersecting ellipses are provided, together with an evaluation of the effectiveness of ellipses in drawing accurate area-proportional Venn diagrams for 3-set data and the characteristics of the data that can be depicted accurately with ellipses

Tactical diagrammatic reasoning

Although automated reasoning with diagrams has been possible for some years, tools for diagrammatic reasoning are generally much less sophisticated than their sentential cousins. The tasks of exploring levels of automation and abstraction in the construction of proofs and of providing explanations of solutions expressed in the proofs remain to be addressed. In this paper we take an interactive proof assistant for Euler diagrams, Speedith, and add tactics to its reasoning engine, providing a level of automation in the construction of proofs. By adding tactics to Speedith's repertoire of inferences, we ease the interaction between the user and the system and capture a higher level explanation of the essence of the proof. We analysed the design options for tactics by using metrics which relate to human readability, such as the number of inferences and the amount of clutter present in diagrams. Thus, in contrast to the normal case with sentential tactics, our tactics are designed to not only prove the theorem, but also to support explanation

Design and Implementation of Small Multiples Matrix-based Visualisation to Monitor and Compare Email Socio-organisational Relationships

One of the fundamental organisational questions is how organisations identify anomalies, monitor and compare email communications between staff-staff or staff-clients or staff-customers relationships on a daily basis. The tenacious and substantial relationships are built by the combination of timely replies, frequent engagement and deep interaction between the individuals. To watchdog this periodically, we need an interactive visualisation tool that can help organisational analysts to reconnect some lost relationships and/or strengthen an existing relationship or in some cases identify inside persons (anomalies). From our point of view, Social Intelligence (SI) in an organisation is a combination of self-, social- and organisational-awareness that will help in managing complex socio-organisational changes and can be interpreted in terms of socio-organisational communication efficacy (that is, one's confidence in one's ability to deal with social and organisational information). We considered a case study, an Enron Organisation Email Scandal, to understand the relationships of staff during various parts of the years and we conducted a workshop study with legal experts to gain insights on how they carry out investigation/analysis with respect to email relationships. The outcomes of the workshop helped us develop a novel small multiples matrix-based visualisation in collaboration with our industrial partner, Red Sift UK, to find anomalies, monitor and compare how email relationships change over time and how it defines the meaning of socio-organisational communication efficacy

Evaluating Visualizations of Sets and Networks that Use Euler Diagrams and Graphs

This paper presents an empirical evaluation of state-of-the-art visualization techniques that combine Euler diagrams and graphs to visualize sets and networks. Focusing on SetNet, Bubble Sets and WebCola – techniques for which there is freely available software – our evaluation reveals that they can inaccurately and ineffectively visualize the data. Inaccuracies include placing vertices in incorrect zones, thus incorrectly conveying the sets in which the represented data items lie. Ineffective properties, which are known to hinder cognition, include drawing Euler diagrams with extra zones or graphs with large numbers of edge crossings. The results demonstrate the need for improved techniques that are more accurate and more effective for end users.The Leverhulme Trus

Accessible reasoning with diagrams: From cognition to automation

High-tech systems are ubiquitous and often safety and se- curity critical: reasoning about their correctness is paramount. Thus, precise modelling and formal reasoning are necessary in order to convey knowledge unambiguously and accurately. Whilst mathematical mod- elling adds great rigour, it is opaque to many stakeholders which leads to errors in data handling, delays in product release, for example. This is a major motivation for the development of diagrammatic approaches to formalisation and reasoning about models of knowledge. In this paper, we present an interactive theorem prover, called iCon, for a highly expressive diagrammatic logic that is capable of modelling OWL 2 ontologies and, thus, has practical relevance. Significantly, this work is the first to design diagrammatic inference rules using insights into what humans find accessible. Specifically, we conducted an experiment about relative cognitive benefits of primitive (small step) and derived (big step) inferences, and use the results to guide the implementation of inference rules in iCon