Network science plays a crucial role in understanding complex systems, with applications ranging from pandemic prediction to drug repurposing and social media analysis. Network visualization, primarily using force-directed layout (FDL) algorithms, is a fundamental step in this process. Popular software packages like Cytoscape and Gephi are widely used, but their reliance on FDL algorithms, with an O(N²) computational cost per iteration, severely limits their applicability to large real-world networks. Visualizing large systems often results in "hairball" layouts, hindering interpretation and insight. This limitation significantly restricts our ability to explore the architecture of large networks, such as protein-protein interaction networks or social media networks, which contain tens of thousands or millions of nodes and links. The current study aims to address this limitation by proposing a novel method to accelerate FDL for large network visualization.

Existing network visualization tools predominantly rely on force-directed layout (FDL) algorithms. These algorithms, inspired by energy minimization in computational chemistry, treat links as springs and employ repulsive forces to prevent node overlap. While effective for smaller networks, the quadratic time complexity (O(N²)) of FDL makes it computationally expensive for large graphs, often leading to poorly interpretable "hairball" visualizations. Previous attempts to improve FDL have focused on algorithmic optimizations, such as Barnes-Hut algorithms to reduce the complexity of repulsive force calculations. However, these methods still face significant challenges when dealing with very large networks.

This research proposes NeuLay, an unsupervised machine learning approach using Graph Neural Networks (GNNs) to accelerate FDL. The core idea is to re-parameterize the energy-based optimization problem behind FDL using GNNs. Two GNN architectures are explored: NodeMLP, which uses a fully connected layer to map high-dimensional node embeddings to the layout's dimensions; and NeuLay, which leverages graph convolutional networks (GCN) to incorporate the graph's structure into the embedding. NeuLay-2, a two-layer GCN architecture, demonstrated optimal performance. The method involves retraining the neural network parameters for each graph layout, as the training process itself is the optimization of the FDL algorithm. The computational complexity is improved by reducing the number of iterations required for convergence, rather than reducing the per-step time complexity. The performance of NeuLay is assessed using two metrics: speed (wall-clock time) and layout quality (potential energy, cluster separation, and link length distribution). The theoretical speedup of NeuLay-2 is analytically derived, linking it to the number of outlier eigenvalues in the adjacency matrix. Experiments are conducted on synthetic networks (Erdős Rényi, Barabási-Albert, Stochastic Block Model, and Random Geometric Graphs) and real-world networks (word association graph, human protein-protein interaction network, Facebook social network, and the Internet at the autonomous system level). For the real-world networks, both the energy and link length distributions are analyzed to evaluate the quality of the layouts. A spatial similarity metric is introduced to measure how well NeuLay-2 separates clusters compared to FDL.

NeuLay-2 consistently outperforms FDL in terms of speed and layout quality across various network types and sizes. Speedups of one to two orders of magnitude are observed, increasing with the network size but decreasing with density. The speedup is particularly pronounced for networks with strong community structures or spatial regularities (e.g., Stochastic Block Model and Random Geometric Graphs), which align with the analytical prediction that speedup increases with the number of outlier eigenvalues. NeuLay-2 frequently converges to lower energy states than FDL, indicating that it avoids local sub-optimal configurations that trap FDL. For large networks, this difference in energy significantly impacts the layout's interpretability, with NeuLay-2 producing more informative layouts that better reveal community structures and display more clustered communities. Specifically, when applied to the Internet graph, NeuLay-2 generated a 3D layout superior to that of FDL. FDL resulted in a "hairball" while NeuLay-2 revealed clear community structures, confirmed through the analysis of community-specific link length distributions and a newly proposed spatial similarity metric which quantitatively supports the visual observations. These findings validate that the speedup is indeed driven by the ability of the neural network to efficiently handle the 'slow modes' (the leading eigenvectors) of the FDL algorithm.

NeuLay's success in accelerating graph layout stems from its ability to effectively handle the slow modes of FDL, which are primarily determined by the leading eigenvectors of the adjacency matrix. By projecting the graph layout onto the top few principal components from the first iteration, NeuLay quickly separates large communities, thereby accelerating convergence. The underlying mechanism is not limited to graph layouts and extends to other energy minimization problems on graphs or graph dynamical processes described by gradient descent. Potential applications include improving the modeling of reaction-diffusion systems, epidemics, cascading failures, and optimizing chip design. The improved layout quality, as measured by lower energy and better community separation, directly translates to more insightful network visualizations, facilitating a better understanding of complex network structures.

The NeuLay algorithm offers a significant advancement in network visualization, providing a faster and more effective approach to laying out large graphs. Its ability to identify and exploit the underlying structure of networks, leading to substantially faster convergence and improved layout quality, makes it a powerful tool for exploring complex systems. Future research could focus on enhancing NeuLay's efficiency by exploiting network symmetries or hierarchical structures and extending it to handle more complex physical networks with curved links.

The current implementation of NeuLay's efficiency is limited by the computational complexity of GNNs, although they are considerably faster than FDL. While short-range repulsive forces are used in the paper, the use of long-range forces might result in even better performance but would make the algorithm computationally intractable. Further research is required to explore these aspects and to optimize NeuLay for exceptionally large graphs.