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Graph and network visualization supports exploration, analysis and communication of relational data arising in many domains: from biological and social networks, to transportation and powergrid systems. With the arrival of AI-based question-answering tools, issues of trustworthiness and explainability of generated answers motivate a greater role for visualization. In the context of graphs, we see the need for visualizations that can convince a critical audience that an assertion about the graph under analysis is valid. The requirements for such representations that convey precisely one specific graph property are quite different from standard network visualization criteria which optimize general aesthetics and readability. In this paper, we aim to provide a comprehensive introduction to visual proofs of graph properties and a foundation for further research in the area. We present a framework that defines what it means to visually prove a graph property. In the process, we introduce the notion of a visual certificate, that is, a specialized faithful graph visualization that leverages the viewer's perception, in particular, pre-attentive processing (e.g. via pop-out effects), to verify a given assertion about the represented graph. We also discuss the relationships between visual complexity, cognitive load and complexity theory, and propose a classification based on visual proof complexity. Finally, we provide examples of visual certificates for problems in different visual proof complexity classes., Comment: Appears in the Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

This paper aims to develop a mathematical foundation to model knitting with graphs. We provide a precise definition for knit objects with a knot theoretic component and propose a simple undirected graph, a simple directed graph, and a directed multigraph model for any arbitrary knit object. Using these models, we propose natural categories related to the complexity of knitting structures. We use these categories to explore the hardness of determining whether a knit object of each class exists for a given graph. We show that while this problem is NP-hard in general, under specific cases, there are linear and polynomial time algorithms which take advantage of unique properties of common knitting techniques. This work aims to bridge the gap between textile arts and graph theory, offering a useful and rigorous framework for analyzing knitting objects using their corresponding graphs and for generating knitting objects from graphs., Comment: 11 pages, 4 tables, 5 figures

Celtic knots are an ancient art form often attributed to Celtic cultures, used to decorate monuments and manuscripts, and to symbolise eternity and interconnectedness. This paper describes the framework CelticGraph to draw graphs as Celtic knots and links. The drawing process raises interesting combinatorial concepts in the theory of circuits in planar graphs. Further, CelticGraph uses a novel algorithm to represent edges as B\'ezier curves, aiming to show each link as a smooth curve with limited curvature., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)

Relational information between different types of entities is often modelled by a multilayer network (MLN) -- a network with subnetworks represented by layers. The layers of an MLN can be arranged in different ways in a visual representation, however, the impact of the arrangement on the readability of the network is an open question. Therefore, we studied this impact for several commonly occurring tasks related to MLN analysis. Additionally, layer arrangements with a dimensionality beyond 2D, which are common in this scenario, motivate the use of stereoscopic displays. We ran a human subject study utilising a Virtual Reality headset to evaluate 2D, 2.5D, and 3D layer arrangements. The study employs six analysis tasks that cover the spectrum of an MLN task taxonomy, from path finding and pattern identification to comparisons between and across layers. We found no clear overall winner. However, we explore the task-to-arrangement space and derive empirical-based recommendations on the effective use of 2D, 2.5D, and 3D layer arrangements for MLNs.

The visualization of a network influences the quality of the mental map that the viewer develops to understand the network. In this study, we investigate the effects of a 3D immersive visualization environment compared to a traditional 2D desktop environment on the comprehension of a network's structure. We compare the two visualization environments using three tasks--interpreting network structure, memorizing a set of nodes, and identifying the structural changes--commonly used for evaluating the quality of a mental map in network visualization. The results show that participants were able to interpret network structure more accurately when viewing the network in an immersive environment, particularly for larger networks. However, we found that 2D visualizations performed better than immersive visualization for tasks that required spatial memory., Comment: IEEE Pacific Visualization Symposium 2020

Extracting significant places or places of interest (POIs) using individuals' spatio-temporal data is of fundamental importance for human mobility analysis. Classical clustering methods have been used in prior work for detecting POIs, but without considering temporal constraints. Usually, the involved parameters for clustering are difficult to determine, e.g., the optimal cluster number in hierarchical clustering. Currently, researchers either choose heuristic values or use spatial distance-based optimization to determine an appropriate parameter set. We argue that existing research does not optimally address temporal information and thus leaves much room for improvement. Considering temporal constraints in human mobility, we introduce an effective clustering approach - namely POI clustering with temporal constraints (PC-TC) - to extract POIs from spatio-temporal data of human mobility. Following human mobility nature in modern society, our approach aims to extract both global POIs (e.g., workplace or university) and local POIs (e.g., library, lab, and canteen). Based on two publicly available datasets including 193 individuals, our evaluation results show that PC-TC has much potential for next place prediction in terms of granularity (i.e., the number of extracted POIs) and predictability.

Background: Systems biology projects and omics technologies have led to a growing number of biochemical pathway reconstructions. However, mathematical models are still most often created de novo, based on reading the literature and processing pathway data manually. Results: To increase the efficiency with which such models can be created, we automatically generated mathematical models from pathway representations using a suite of freely available software. We produced models that combine data from KEGG PATHWAY, BioCarta, MetaCyc and SABIO-RK; According to the source data, three types of models are provided: kinetic, logical and constraint-based. All models are encoded using SBML Core and Qual packages, and available through BioModels Database. Each model contains the list of participants, the interactions, and the relevant mathematical constructs, but, in most cases, no meaningful parameter values. Most models are also available as easy to understand graphical SBGN maps. Conclusions: to date, the project has resulted in more than 140000 models freely available. We believe this resource can tremendously accelerate the development of mathematical models by providing initial starting points ready for parametrization., Comment: 29 pages, 8 figures