The spring bounces back: introducing the strain elevation tension spring embedding algorithm for network representation

This paper introduces the strain elevation tension spring embedding (SETSe) algorithm. SETSe is a novel graph embedding method that uses a physical model to project feature-rich networks onto a manifold with semi-Euclidean properties. Due to its method, SETSe avoids the tractability issues faced by traditional force-directed graphs, having an iteration time and memory complexity that is linear to the number of edges in the network. SETSe is unusual as an embedding method as it does not reduce dimensionality or explicitly attempt to place similar nodes close together in the embedded space. Despite this, the algorithm outperforms five common graph embedding algorithms, on graph classification and node classification tasks, in low-dimensional space. The algorithm is also used to embed 100 social networks ranging in size from 700 to over 40,000 nodes and up to 1.5 million edges. The social network embeddings show that SETSe provides a more expressive alternative to the popular assortativity metric and that even on large complex networks, SETSe’s classification ability outperforms the naive baseline and the other embedding methods in low-dimensional representation. SETSe is a fast and flexible unsupervised embedding algorithm that integrates node attributes and graph topology to produce interpretable results

Force-directed embedding of scale-free networks in the hyperbolic plane

Force-directed drawing algorithms are the most commonly used approach to visualize networks. While they are usually very robust, the performance of Euclidean spring embedders decreases if the graph exhibits the high level of heterogeneity that typically occurs in scale-free real-world networks. As heterogeneity naturally emerges from hyperbolic geometry (in fact, scale-free networks are often perceived to have an underlying hyperbolic geometry), it is natural to embed them into the hyperbolic plane instead. Previous techniques that produce hyperbolic embeddings usually make assumptions about the given network, which (if not met) impairs the quality of the embedding. It is still an open problem to adapt force-directed embedding algorithms to make use of the heterogeneity of the hyperbolic plane, while also preserving their robustness. We identify fundamental differences between the behavior of spring embedders in Euclidean and hyperbolic space, and adapt the technique to take advantage of the heterogeneity of the hyperbolic plane

Fast Spherical Drawing of Triangulations: An Experimental Study of Graph Drawing Tools

We consider the problem of computing a spherical crossing-free geodesic drawing of a planar graph: this problem, as well as the closely related spherical parameterization problem, has attracted a lot of attention in the last two decades both in theory and in practice, motivated by a number of applications ranging from texture mapping to mesh remeshing and morphing. Our main concern is to design and implement a linear time algorithm for the computation of spherical drawings provided with theoretical guarantees. While not being aesthetically pleasing, our method is extremely fast and can be used as initial placer for spherical iterative methods and spring embedders. We provide experimental comparison with initial placers based on planar Tutte parameterization. Finally we explore the use of spherical drawings as initial layouts for (Euclidean) spring embedders: experimental evidence shows that this greatly helps to untangle the layout and to reach better local minima

The spring bounces back: Introducing the Strain Elevation Tension Spring embedding algorithm for network representation

This paper introduces the Strain Elevation Tension Spring embedding (SETSe) algorithm, a graph embedding method that uses a physics model to create node and edge embeddings in undirected attribute networks. Using a low-dimensional representation, SETSe is able to differentiate between graphs that are designed to appear identical using standard network metrics such as number of nodes, number of edges and assortativity. The embeddings generated position the nodes such that sub-classes, hidden during the embedding process, are linearly separable, due to the way they connect to the rest of the network. SETSe outperforms five other common graph embedding methods on both graph differentiation and sub-class identification. The technique is applied to social network data, showing its advantages over assortativity as well as SETSe's ability to quantify network structure and predict node type. The algorithm has a convergence complexity of around $\mathcal{O}(n^2)$, and the iteration speed is linear ($\mathcal{O}(n)$), as is memory complexity. Overall, SETSe is a fast, flexible framework for a variety of network and graph tasks, providing analytical insight and simple visualisation for complex systems.Comment: 27 pages; 7000 word

Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane

Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of Omega(n^2). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth

Strain Elevation Tension Spring embedding and Cascading failures on the power-grid

Understanding the dynamics and properties of networks is of great importance in our highly connected data-driven society. When the networks relate to infrastructure, such understanding can have a substantial impact on public welfare. As such, there is a need for algorithms that can provide insights into the observable and latent properties of these structures. This thesis presents a novel embedding algorithm: the Strain Elevation Tension Spring embedding (SETSe), as a method of understanding complex networks. The algorithm is a deterministic physics model that incorporates both node and edge features into the final embedding. SETSe distinguishes itself from most embeddings methods by not having a loss function in the conventional sense and by not trying to place similar nodes close together. Instead, SETSe acts as a smoothing function for node features across the network topology. This approach produces embeddings that are intuitive and interpretable. In this thesis, I demonstrate how SETSe outperforms alternative embedding methods on node level and graph level tasks using networks made from stochastic block models and social networks with over 40,000 nodes and over 1 million edges. I also highlight a weakness of traditional methods to analysing cascading failures on power grids and demonstrate that SETSe is not susceptible to such issues. I then show how SETSe can be used as a measure of robustness in addition to providing a means to create interpretable maps in the geographical space given its smoothing embedding method. The framework has been made widely available through two open source R packages contributions, 1) the implementation of SETSe ("rsetse" on CRAN), and 2) a package for analysing cascading failures on power grids

A Sparse Stress Model

Force-directed layout methods constitute the most common approach to draw general graphs. Among them, stress minimization produces layouts of comparatively high quality but also imposes comparatively high computational demands. We propose a speed-up method based on the aggregation of terms in the objective function. It is akin to aggregate repulsion from far-away nodes during spring embedding but transfers the idea from the layout space into a preprocessing phase. An initial experimental study informs a method to select representatives, and subsequent more extensive experiments indicate that our method yields better approximations of minimum-stress layouts in less time than related methods.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016