# Detailed Examples of Figure Preparation in the Two Most Common Graph Layouts
## 1 Introduction  
In the realm of engineering, where complexity and interconnectivity reign supreme, the need to comprehend and analyze vast networks of relationships is more crucial than ever. Graphs represent an ingenious and transformative tool that has found its way into various branches of engineering, revolutionizing the ways that we perceive and solve problems. From civil and mechanical engineering to computer science and beyond, graphs have become an indispensable asset, propelling innovation, streamlining processes, and offering hidden insights that were once beyond the grasp of traditional methodologies. Graph theory, with its roots dating back to the early 18 th century, has metamorphosed into a dynamic and multifaceted discipline with a wide range of applications in engineering. At its core, a graph represents a collection of nodes (vertices) connected by edges (links) that embody relationships, dependencies, or interactions between them. With this seemingly simple concept, engineers have gained the power to capture and model complex systems, enabling them to tackle challenges of immense intricacy in an elegant and systematic manner. Graphs can be utilized to analyze transportation networks, optimize supply chains, design efficient power grids, or simulate fluid dynamics. There are many graph types (such as random geometric graphs, Erdos-Renyi graphs, small-world graphs, scale-free graphs, etc.) that find application in various domains (e.g., wireless sensor networks, bioinformatics, social networks, neural networks, etc.). For some examples of graph applications, see, e.g., [1, 2, 3, 4]. The versatility of graphs lies not only in their ability torepresent physical structures but also in their capacity to illuminate abstract relationships, fostering a deeper understanding of the world around us. Furthermore, as technology continues to advance at an exponential pace, the volume of data generated in engineering projects has reached unprecedented levels. In such a data-rich environment, graphs have emerged as a key tool in organizing, querying, and extracting meaningful insights from this wealth of information. Leveraging powerful graph algorithms, engineers can identify patterns, detect anomalies, and make informed decisions that result in optimized designs, improved safety measures, and, ultimately, a more sustainable future. In this context, graph layouts have emerged as a powerful tool, becoming a cornerstone in the visual representation and analysis of complex networks. At its essence, a graph layout aims to translate the abstract relationships between nodes and edges into meaningful spatial arrangements, making it easier for researchers and analysts to identify crucial insights, detect anomalies, and make informed choices. One of the most compelling aspects of graph layouts is their versatility in adapting to a vast array of network structures. Whether this is a sprawling social media network with millions of users, a complex supply chain that spans continents, or a microscopic web of molecular interactions within a cell, graph layouts provide a coherent representation that transcends scale and complexity. There are various algorithms for graph drawing. The simplest ones are random layouts, where the coordinates for the vertices of the graph are chosen randomly, or circular layouts, where all vertices are placed evenly on a circle. This type of visualization does not reveal any hidden graph features. Therefore, many more sophisticated algorithms have been proposed. These include, e.g., Kamada-Kwai [5], Fruchterman-Reingold [6], ForceAtlas2 [7], and Open Ord [8]. Each of these has a certain specificity that allows it to emphasize different features and limitations of the graph. Among the algorithms used to build graph drawings, there are certain classes. Some of them, for example, emphasize transparency, while others are better for large graphs (with millions of nodes).