Graph Drawing Algorithms Research Articles

SmartGD: A GAN-Based Graph Drawing Framework for Diverse Aesthetic Goals.

While a multitude of studies have been conducted on graph drawing, many existing methods only focus on optimizing a single aesthetic aspect of graph layouts, which can lead to sub-optimal results. There are a few existing methods that have attempted to develop a flexible solution for optimizing different aesthetic aspects measured by different aesthetic criteria. Furthermore, thanks to the significant advance in deep learning techniques, several deep learning-based layout methods were proposed recently. These methods have demonstrated the advantages of deep learning approaches for graph drawing. However, none of these existing methods can be directly applied to optimizing non-differentiable criteria without special accommodation. In this work, we propose a novel Generative Adversarial Network (GAN) based deep learning framework for graph drawing, called SmartGD, which can optimize different quantitative aesthetic goals, regardless of their differentiability. To demonstrate the effectiveness and efficiency of SmartGD, we conducted experiments on minimizing stress, minimizing edge crossing, maximizing crossing angle, maximizing shape-based metrics, and a combination of multiple aesthetics. Compared with several popular graph drawing algorithms, the experimental results show that SmartGD achieves good performance both quantitatively and qualitatively.

Automorphism Faithfulness Metrics for Symmetric Graph Drawings.

In this article, we present new quality metrics for symmetric graph drawing based on group theory. Roughly speaking, the new metrics are faithfulness metrics, i.e., they measure how faithfully a drawing of a graph displays the ground truth (i.e., geometric automorphisms) of the graph as symmetries. More specifically, we introduce two types of automorphism faithfulness metrics for displaying: (1) a single geometric automorphism as a symmetry (axial or rotational), and (2) a group of geometric automorphisms (cyclic or dihedral). We present algorithms to compute the automorphism faithfulness metrics in O(n logn) time. Moreover, we also present efficient algorithms to detect exact symmetries in a graph drawing. We then validate our automorphism faithfulness metrics using deformation experiments. Finally, we use the metrics to evaluate existing graph drawing algorithms to compare how faithfully they display geometric automorphisms of a graph as symmetries.

Relating planar graph drawings to planar satisfiability problems

A SAT graph G(Φ) of a satisfiability instance Φ consists of a vertex for each clause and a vertex for each variable, where there exists an edge between a clause vertex and a variable vertex if and only if the variable or its negation appears in that clause. Many satisfiability problems, which are NP-hard, become polynomial-time solvable when the SAT graph is restricted to satisfy some graph properties. A rich body of research attempts to narrow down the boundary between the NP-hardness and polynomial-time solvability of various satisfiability problems. In this paper, we examine planar satisfiability problems and leverage planar graph drawing algorithms to improve our understanding of these problems. A rich body of graph drawing algorithms exists to check whether a planar graph admits a drawing that satisfies certain drawing aesthetics. We show how the existing graph drawing knowledge could be used to establish sufficient conditions for a SAT instance to always be satisfiable and give algorithms to efficiently find a satisfying truth assignment. In some cases, our algorithm can find a truth assignment by setting a small number of variables to true, which relates to the satisfiability variants that seek to minimize the number of ones.

$st$-Orientations with Few Transitive Edges

The problem of orienting the edges of an undirected graph such that the resulting digraph is acyclic and has a single source $s$ and a single sink $t$ has a long tradition in graph theory and is central to many graph drawing algorithms. Such an orientation is called an $st$-orientation. We address the problem of computing $st$-orientations of undirected graphs with the minimum number of transitive edges. We prove that the problem is NP-hard in the general case. For planar graphs we describe an ILP (Integer Linear Programming) model that is fast in practice, namely it takes on average less than 1 second for graphs with up to 100 vertices, and about 10 seconds for larger instances with up to 1000 vertices. We experimentally show that optimum solutions significantly reduce (35% on average) the number of transitive edges with respect to unconstrained $st$-orientations computed via classical $st$-numbering algorithms. Moreover, focusing on popular graph drawing algorithms that apply an $st$-orientation as a preliminary step, we show that reducing the number of transitive edges leads to drawings that are much more compact (with an improvement between 30% and 50% for most of the instances).

Tightening Curves on Surfaces Monotonically with Applications

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of De Graaf and Schrijver [ J. Comb. Theory Ser. B , 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound, we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k -terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and Δ Y -transformations for arbitrary integer k . Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k . Our algorithm makes use of the connection between electrical transformations and homotopy moves and thus solves the problem in a unified fashion.

Event‐based Dynamic Graph Drawing without the Agonizing Pain

AbstractTemporal networks can naturally model real‐world complex phenomena such as contact networks, information dissemination and physical proximity. However, nodes and edges bear real‐time coordinates, making it difficult to organize them into discrete timeslices, without a loss of temporal information due to projection. Event‐based dynamic graph drawing rejects the notion of a timeslice and allows each node and edge to retain its own real‐valued time coordinate. While existing work has demonstrated clear advantages for this approach, they come at a running time cost. We investigate the problem of accelerating event‐based layout to make it more competitive with existing layout techniques. In this paper, we describe the design, implementation and experimental evaluation of MultiDynNoS, the first multi‐level event‐based graph layout algorithm. We consider three operators for coarsening and placement, inspired by Walshaw, GRIP and FM3, which we couple with an event‐based graph drawing algorithm. We also propose two extensions to the core algorithm: AutoTau and Bend Transfer. We perform two experiments: first, we compare MultiDynNoS variants to existing state‐of‐the‐art dynamic graph layout approaches; second, we investigate the impact of each of the proposed algorithm extensions. MultiDynNoS proves to be competitive with existing approaches, and the proposed extensions achieve their design goals and contribute in opening new research directions.

Granger causality procedeture to diagnosis and failture in industrial systems

Industrial process supervision is an important subject now days due to the increased requirement for safer processes for operators and effective for companies. Control loops affected by disturbs, are grouped with PCA, based on their increased variability and the causal relationships between them are detected via Granger causality. A graph drawing algorithm allows indicating the source of the disturbance. The procedure is applied to data from a simulated chemical process CSTR. The proposed procedeture correctly indicated the sources of disturbances.

Improvement of Fuzzy Graph Drawing Using Partition Tree

Obtaining useful information from ambiguous information is a necessity in various fields. Ambiguous information can be handled quantitatively by using fuzzy theory, and representing it in an easy-to-understand manner is critical. One solution is to visualize an ambiguous relationship by using fuzzy graph representation, which has the essential characteristic of expressing variable relationships in between its nodes. We previously proposed an algorithm to draw intelligible and comprehensive fuzzy graphs. This study describes an improved drawing method for that graph drawing algorithm. As a result, highly related nodes were arranged closer to one another, and the display area was reduced. This method can be used as an effective means of expressing the results of ambiguous information analysis.

Multicriteria Scalable Graph Drawing via Stochastic Gradient Descent, (SGD)2.

Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Multicriteria Scalable Graph Drawing via Stochastic Gradient Descent, (SGD)2, that can handle multiple readability criteria. (SGD)2 can optimize any criterion that can be described by a differentiable function. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., node resolution, angular resolution, aspect ratio). The approach is scalable and can handle large graphs. A variation of the underlying approach can also be used to optimize many desirable properties in planar graphs, while maintaining planarity. Finally, we provide quantitative and qualitative evidence of the effectiveness of (SGD)2: we analyze the interactions between criteria, measure the quality of layouts generated from (SGD)2 as well as the runtime behavior, and analyze the impact of sample sizes. The source code is available on github and we also provide an interactive demo for small graphs.

DeepGD: A Deep Learning Framework for Graph Drawing Using GNN.

In the past decades, many graph drawing techniques have been proposed for generating aesthetically pleasing graph layouts. However, it remains a challenging task since different layout methods tend to highlight different characteristics of the graphs. Recently, studies on deep-learning-based graph drawing algorithms have emerged but they are often not generalizable to arbitrary graphs without retraining. In this article, we propose a Convolutional-Graph-Neural-Network-based deep learning framework, DeepGD, which can draw arbitrary graphs once trained. It attempts to generate layouts by compromising among multiple prespecified aesthetics considering a good graph layout usually complies with multiple aesthetics simultaneously. In order to balance the tradeoff, we propose two adaptive training strategies, which adjust the weight factor of each aesthetic dynamically during training. The quantitative and qualitative assessment of DeepGD demonstrates that it is capable of drawing arbitrary graphs effectively, while being flexible at accommodating different aesthetic criteria.

BC tree-based spectral sampling for big complex network visualization

Graph sampling methods have been used to reduce the size and complexity of big complex networks for graph mining and visualization. However, existing graph sampling methods often fail to preserve the connectivity and important structures of the original graph. This paper introduces a new divide and conquer approach to spectral graph sampling based on graph connectivity, called the BC Tree (i.e., decomposition of a connected graph into biconnected components) and spectral sparsification. Specifically, we present two methods, spectral vertex sampling BC_SV and spectral edge sampling BC_SS by computing effective resistance values of vertices and edges for each connected component. Furthermore, we present DBC_SS and DBC_GD, graph connectivity-based distributed algorithms for spectral sparsification and graph drawing respectively, aiming to further improve the runtime efficiency of spectral sparsification and graph drawing by integrating connectivity-based graph decomposition and distributed computing. Experimental results demonstrate that BC_SV and BC_SS are significantly faster than previous spectral graph sampling methods while preserving the same sampling quality. DBC_SS and DBC_GD obtain further significant runtime improvement over sequential approaches, and DBC_GD further achieves significant improvements in quality metrics over sequential graph drawing layouts.

A coarsening method for bipartite networks via weight-constrained label propagation

A multilevel method is a scalable strategy to solve optimization problems in large bipartite networks, which operates in three stages. Initially the input network is iteratively coarsened into a hierarchy of gradually smaller networks. Coarsening implies in collapsing vertices into so-called super-vertices which inherit properties of their originating vertices. An initial solution is obtained executing the target algorithm in the coarsest network. Finally, this solution is successively projected back over the inverse sequence of coarsened networks, up to the initial one, yielding an approximate final solution. Despite its potential applicability, the strategy faces several theoretical and practical limitations. Coarsening is usually attained following a user-defined policy to match vertices pairwise. However, the network reduction process is extremely slow and may yield degraded solutions due to propagation of poor matches. Additionally, proper parameterization of coarsening algorithms is difficult, as well as ensuring the super-vertices preserve the relevant properties. We address these issues with a near-linear complexity coarsening strategy based on weight-constrained label propagation. Our strategy collapses groups of vertices, rather than pairs, yielding faster and more extensive network reduction. Moreover, users may specify the desired size of the coarsest network and control super-vertex weights. The applicability of our solution is illustrated in multiple scenarios, namely: multilevel implementation of an existing high-cost community detection algorithm; as a direct community detection algorithm; finally, network visualization, in connection with force-directed graph drawing algorithms. Results provide empirical evidence on the potential of our proposal to foster novel applications of the multilevel method in bipartite networks.

Spectrum-preserving sparsification for visualization of big graphs

We present a novel spectrum-preserving sparsification algorithm for visualizing big graph data. Although spectral methods have many advantages, the high memory and computation costs due to the involved Laplacian eigenvalue problems could immediately hinder their applications in big graph analytics. In this paper, we introduce a practically efficient, nearly-linear time spectral sparsification algorithm for tackling real-world big graph data. Besides spectral sparsification, we further propose a node reduction scheme based on intrinsic spectral graph properties to allow more aggressive, level-of-detail simplification. To enable effective visual exploration of the resulting spectrally sparsified graphs, we implement spectral clustering and edge bundling. Our framework does not depend on a particular graph layout and can be integrated into different graph drawing algorithms. We experiment with publicly available graph data of different sizes and characteristics to demonstrate the efficiency and effectiveness of our approach. To further verify our solution, we quantitatively compare our method against different graph simplification solutions using a proxy quality metric and statistical properties of the graphs.

Convexity-increasing morphs of planar graphs

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal.To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.

DiSTruct v1.0: generating biomolecular structures from distance constraints.

The distance geometry problem is often encountered in molecular biology and the life sciences at large, as a host of experimental methods produce ambiguous and noisy distance data. In this note, we present diSTruct; an adaptation of the generic MaxEnt-Stress graph drawing algorithm to the domain of biological macromolecules. diSTruct is fast, provides reliable structural models even from incomplete or noisy distance data and integrates access to graph analysis tools. diSTruct is written in C++, Cython and Python 3. It is available from https://github.com/KIT-MBS/distruct.git or in the Python package index under the MIT license. Supplementary data are available at Bioinformatics online.

An Experimental Study on the Ply Number of Straight-line Drawings

The ply number of a drawing is a new criterion of interest for graph drawing. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the longest edge incident to that vertex. This paper reports the results of an extensive experimental study that attempts to estimate correlations between the ply number and other aesthetic quality metrics for a graph layout, such as stress, edge-length uniformity, and edge crossings. We also investigate the performance of several graph drawing algorithms in terms of ply number, and provide new insights into the theoretical gap between lower and upper bounds on the ply number of $k$-ary trees.

Applying Graph Centrality Metrics in Visual Analytics of Scientific Standard Datasets

Graphs are often used to model data with a relational structure and graphs are usually visualised into node-link diagrams for a better understanding of the underlying data. Node-link diagrams represent not only data entries in a graph, but also the relations among the data entries. Further, many graph drawing algorithms and graph centrality metrics have been successfully applied in visual analytics of various graph datasets, yet little attention has been paid to analytics of scientific standard data. This study attempts to adopt graph drawing methods (force-directed algorithms) to visualise scientific standard data and provide information with importance ‘ranking’ based on graph centrality metrics such as Weighted Degree, PageRank, Eigenvector, Betweenness and Closeness factors. The outcomes show that our method can produce clear graph layouts of scientific standard for visual analytics, along with the importance ‘ranking’ factors (represent via node colour, size etc.). Our method may assist users with tracking various relationships while understanding scientific standards with fewer relation issues (missing/wrong connection etc.) through focusing on higher priority standards.

Event-Based Dynamic Graph Visualisation.

Dynamic graph drawing algorithms take as input a series of timeslices that standard, force-directed algorithms can exploit to compute a layout. However, often dynamic graphs are expressed as a series of events where the nodes and edges have real coordinates along the time dimension that are not confined to discrete timeslices. Current techniques for dynamic graph drawing impose a set of timeslices on this event-based data in order to draw the dynamic graph, but it is unclear how many timeslices should be selected: too many timeslices slows the computation of the layout, while too few timeslices obscures important temporal features, such as causality. To address these limitations, we introduce a novel model for drawing event-based dynamic graphs and the first dynamic graph drawing algorithm, DynNoSlice, that is capable of drawing dynamic graphs in this model. DynNoSlice is an offline, force-directed algorithm that draws event-based, dynamic graphs in the space-time cube (2D+time). We also present a method to extract representative small multiples from the space-time cube. To demonstrate the advantages of our approach, DynNoSlice is compared with state-of-the-art timeslicing methods using a metrics-based experiment. Finally, we present case studies of event-based dynamic data visualised with the new model and algorithm.

Testing SensoGraph, a geometric approach for fast sensory evaluation

This paper introduces SensoGraph, a novel approach for fast sensory evaluation using two-dimensional geometric techniques. In the tasting sessions, the assessors follow their own criteria to place samples on a tablecloth, according to the similarity between samples. In order to analyse the data collected, first a geometric clustering is performed to each tablecloth, extracting connections between the samples. Then, these connections are used to construct a global similarity matrix. Finally, a graph drawing algorithm is used to obtain a 2D consensus graphic, which reflects the global opinion of the panel by (1) positioning closer those samples that have been globally perceived as similar and (2) showing the strength of the connections between samples. The proposal is validated by performing four tasting sessions, with three types of panels tasting different wines, and by developing a new software to implement the proposed techniques. The results obtained show that the graphics provide similar positionings of the samples as the consensus maps obtained by multiple factor analysis (MFA), further providing extra information about connections between samples, not present in any previous method. The main conclusion is that the use of geometric techniques provides information complementary to MFA, and of a different type. Finally, the method proposed is computationally able to manage a significantly larger number of assessors than MFA, which can be useful for the comparison of pictures by a huge number of consumers, via the Internet.

EClerize: A customized force-directed graph drawing algorithm for biological graphs with EC attributes.

Visualizing large-scale data produced by the high throughput experiments as a biological graph leads to better understanding and analysis. This study describes a customized force-directed layout algorithm, EClerize, for biological graphs that represent pathways in which the nodes are associated with Enzyme Commission (EC) attributes. The nodes with the same EC class numbers are treated as members of the same cluster. Positions of nodes are then determined based on both the biological similarity and the connection structure. EClerize minimizes the intra-cluster distance, that is the distance between the nodes of the same EC cluster and maximizes the inter-cluster distance, that is the distance between two distinct EC clusters. EClerize is tested on a number of biological pathways and the improvement brought in is presented with respect to the original algorithm. EClerize is available as a plug-in to Cytoscape ( http://apps.cytoscape.org/apps/eclerize ).