Embedding Graphs under Centrality Constraints for Network Visualization

Visual rendering of graphs is a key task in the mapping of complex network data. Although most graph drawing algorithms emphasize aesthetic appeal, certain applications such as travel-time maps place more importance on visualization of structural network properties. The present paper advocates two graph embedding approaches with centrality considerations to comply with node hierarchy. The problem is formulated first as one of constrained multi-dimensional scaling (MDS), and it is solved via block coordinate descent iterations with successive approximations and guaranteed convergence to a KKT point. In addition, a regularization term enforcing graph smoothness is incorporated with the goal of reducing edge crossings. A second approach leverages the locally-linear embedding (LLE) algorithm which assumes that the graph encodes data sampled from a low-dimensional manifold. Closed-form solutions to the resulting centrality-constrained optimization problems are determined yielding meaningful embeddings. Experimental results demonstrate the efficacy of both approaches, especially for visualizing large networks on the order of thousands of nodes.Comment: Submitted to IEEE Transactions on Visualization and Computer Graphic

Computing Hive Plots: A Combinatorial Framework

Hive plots are a graph visualization style placing vertices on a set of radial axes emanating from a common center and drawing edges as smooth curves connecting their respective endpoints. In previous work on hive plots, assignment to an axis and vertex positions on each axis were determined based on selected vertex attributes and the order of axes was prespecified. Here, we present a new framework focusing on combinatorial aspects of these drawings to extend the original hive plot idea and optimize visual properties such as the total edge length and the number of edge crossings in the resulting hive plots. Our framework comprises three steps: (1) partition the vertices into multiple groups, each corresponding to an axis of the hive plot; (2) optimize the cyclic axis order to bring more strongly connected groups near each other; (3) optimize the vertex ordering on each axis to minimize edge crossings. Each of the three steps is related to a well-studied, but NP-complete computational problem. We combine and adapt suitable algorithmic approaches, implement them as an instantiation of our framework and show in a case study how it can be applied in a practical setting. Furthermore, we conduct computational experiments to gain further insights regarding algorithmic choices of the framework. The code of the implementation and a prototype web application can be found on OSF.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Complexity Analysis of Balloon Drawing for Rooted Trees

In a balloon drawing of a tree, all the children under the same parent are placed on the circumference of the circle centered at their parent, and the radius of the circle centered at each node along any path from the root reflects the number of descendants associated with the node. Among various styles of tree drawings reported in the literature, the balloon drawing enjoys a desirable feature of displaying tree structures in a rather balanced fashion. For each internal node in a balloon drawing, the ray from the node to each of its children divides the wedge accommodating the subtree rooted at the child into two sub-wedges. Depending on whether the two sub-wedge angles are required to be identical or not, a balloon drawing can further be divided into two types: even sub-wedge and uneven sub-wedge types. In the most general case, for any internal node in the tree there are two dimensions of freedom that affect the quality of a balloon drawing: (1) altering the order in which the children of the node appear in the drawing, and (2) for the subtree rooted at each child of the node, flipping the two sub-wedges of the subtree. In this paper, we give a comprehensive complexity analysis for optimizing balloon drawings of rooted trees with respect to angular resolution, aspect ratio and standard deviation of angles under various drawing cases depending on whether the tree is of even or uneven sub-wedge type and whether (1) and (2) above are allowed. It turns out that some are NP-complete while others can be solved in polynomial time. We also derive approximation algorithms for those that are intractable in general

Anisotropic Radial Layout for Visualizing Centrality and Structure in Graphs

This paper presents a novel method for layout of undirected graphs, where nodes (vertices) are constrained to lie on a set of nested, simple, closed curves. Such a layout is useful to simultaneously display the structural centrality and vertex distance information for graphs in many domains, including social networks. Closed curves are a more general constraint than the previously proposed circles, and afford our method more flexibility to preserve vertex relationships compared to existing radial layout methods. The proposed approach modifies the multidimensional scaling (MDS) stress to include the estimation of a vertex depth or centrality field as well as a term that penalizes discord between structural centrality of vertices and their alignment with this carefully estimated field. We also propose a visualization strategy for the proposed layout and demonstrate its effectiveness using three social network datasets.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

The Perception of Graph Properties In Graph Layouts

abstract: When looking at drawings of graphs, questions about graph density, community structures, local clustering and other graph properties may be of critical importance for analysis. While graph layout algorithms have focused on minimizing edge crossing, symmetry, and other such layout properties, there is not much known about how these algorithms relate to a user’s ability to perceive graph properties for a given graph layout. This study applies previously established methodologies for perceptual analysis to identify which graph drawing layout will help the user best perceive a particular graph property. A large scale (n = 588) crowdsourced experiment is conducted to investigate whether the perception of two graph properties (graph density and average local clustering coefficient) can be modeled using Weber’s law. Three graph layout algorithms from three representative classes (Force Directed - FD, Circular, and Multi-Dimensional Scaling - MDS) are studied, and the results of this experiment establish the precision of judgment for these graph layouts and properties. The findings demonstrate that the perception of graph density can be modeled with Weber’s law. Furthermore, the perception of the average clustering coefficient can be modeled as an inverse of Weber’s law, and the MDS layout showed a significantly different precision of judgment than the FD layout.Dissertation/ThesisMasters Thesis Computer Science 201

Visualizing Geophylogenies - Internal and External Labeling with Phylogenetic Tree Constraints

A geophylogeny is a phylogenetic tree where each leaf (biological taxon) has an associated geographic location (site). To clearly visualize a geophylogeny, the tree is typically represented as a crossing-free drawing next to a map. The correspondence between the taxa and the sites is either shown with matching labels on the map (internal labeling) or with leaders that connect each site to the corresponding leaf of the tree (external labeling). In both cases, a good order of the leaves is paramount for understanding the association between sites and taxa. We define several quality measures for internal labeling and give an efficient algorithm for optimizing them. In contrast, minimizing the number of leader crossings in an external labeling is NP-hard. We show nonetheless that optimal solutions can be found in a matter of seconds on realistic instances using integer linear programming. Finally, we provide several efficient heuristic algorithms and experimentally show them to be near optimal on real-world and synthetic instances