NetLSD: Hearing the Shape of a Graph (1805.10712v2)

Abstract: Comparison among graphs is ubiquitous in graph analytics. However, it is a hard task in terms of the expressiveness of the employed similarity measure and the efficiency of its computation. Ideally, graph comparison should be invariant to the order of nodes and the sizes of compared graphs, adaptive to the scale of graph patterns, and scalable. Unfortunately, these properties have not been addressed together. Graph comparisons still rely on direct approaches, graph kernels, or representation-based methods, which are all inefficient and impractical for large graph collections. In this paper, we propose the Network Laplacian Spectral Descriptor (NetLSD): the first, to our knowledge, permutation- and size-invariant, scale-adaptive, and efficiently computable graph representation method that allows for straightforward comparisons of large graphs. NetLSD extracts a compact signature that inherits the formal properties of the Laplacian spectrum, specifically its heat or wave kernel; thus, it hears the shape of a graph. Our evaluation on a variety of real-world graphs demonstrates that it outperforms previous works in both expressiveness and efficiency.

• The paper introduces NetLSD, a novel method that uses the Laplacian spectrum to create heat trace signatures for expressive graph comparison.
• The paper demonstrates efficiency and scalability by employing Taylor expansions and eigenspectrum approximations to handle graphs with millions of nodes.
• The paper ensures size-invariance and captures both local and global structures through normalization using complete and empty reference graphs.

An Overview of NetLSD: Spectral Graph Comparison

The paper "NetLSD: Hearing the Shape of a Graph" introduces a novel method for comparing graphs based on spectral representations. The authors focus on addressing three critical challenges inherent in graph comparison: permutation-invariance, scale-adaptivity, and size-invariance. Existing graph comparison methods, such as graph kernels and edit distances, often falter in at least one of these aspects, especially when scalability is also a concern. NetLSD, which stands for Network Laplacian Spectral Descriptor, leverages the properties of the Laplacian spectrum to provide an efficient and expressive method for comparing large collections of graphs across different scales.

Key Contributions

• Spectral Representation: NetLSD employs the Laplacian spectrum of the graph to generate a heat trace signature. This signature is computed over multiple scales, providing a rich representation of the graph's structure. The use of spectral properties makes the method permutation-invariant, as isomorphic graphs will naturally have the same spectrum.
• Efficiency and Scalability: The method uses a combination of Taylor expansions and eigenspectrum approximations to maintain computational efficiency. This allows NetLSD to handle graphs with millions of nodes, achieving significant scalability without sacrificing accuracy.
• Expressiveness: The heat trace signature captures both local and global graph properties, making NetLSD adaptive to various scales of structural patterns. This is crucial for tasks like community detection, where understanding both micro and macro-structural traits is essential.
• Size-Invariance through Normalization: To tackle size discrepancies among graphs, the authors propose a normalization scheme using complete and empty reference graphs. This ensures that the comparison focuses on structural features rather than merely the number of nodes.

Experimental Evaluation

NetLSD was evaluated against other graph comparison methods, including NetSimile and FGSD, across several datasets from bioinformatics and social networks. The authors used the area under the ROC curve (AUC) and accuracy in a 1-nearest-neighbor (1-NN) classification task to benchmark performance. Notably, NetLSD excelled in distinguishing between real and synthetic graphs and exhibited strong performance in graph classification tasks.

In community detection, NetLSD demonstrated its ability to capture both local and global structural characteristics, outperforming competing methods in distinguishing graphs with and without community structure. The method's effectiveness was further validated in experiments using graphs of varying size distributions, emphasizing its robustness to size-based variations.

Implications and Future Directions

The introduction of NetLSD opens several avenues for further exploration. Due to its inherent scalability and efficiency, it is well-suited for applications involving large-scale graph databases found in social networks and scientific domains. Moreover, its ability to offer insights into multi-scale patterns could be leveraged in graph-based machine learning tasks, such as anomaly detection and graph embedding.

The use of spectral methods in graph analysis, as exemplified by NetLSD, can inspire future work integrating spectral approaches with graph neural networks, potentially enhancing both interpretability and generalization capabilities. Additionally, exploring other spectral properties or incorporating domain-specific adaptations could further refine and extend the applicability of techniques like NetLSD.

In summary, the paper presents a solid advancement in spectral graph comparison. NetLSD offers a comprehensive framework that balances expressiveness, efficiency, and invariance, effectively addressing many limitations of existing graph analytic techniques. This holds substantial promise for both theoretical advancements and practical deployments in various domains.