[Paper Review] Graph Distance from the Topological View of Non-backtracking Cycles

Leo Torres, Pablo Suárez–Serrato+1 more|arXiv (Cornell University)|Jul 20, 2018

Topological and Geometric Data Analysis41 references4 citations

TL;DR

This paper introduces the Truncated Non-Backtracking Spectral Distance (TNBSD), a principled, interpretable, and efficient graph distance measure based on the length spectrum of non-backtracking cycles in undirected, unweighted graphs. By leveraging the eigenvalues of the non-backtracking matrix and their topological interpretation via free homotopy classes, TNBSD effectively discriminates between real and synthetic networks, with strong performance in visualization, clustering, and anomaly detection.

ABSTRACT

Whether comparing networks to each other or to random expectation, measuring dissimilarity is essential to understanding the complex phenomena under study. However, determining the structural dissimilarity between networks is an ill-defined problem, as there is no canonical way to compare two networks. Indeed, many of the existing approaches for network comparison differ in their heuristics, efficiency, interpretability, and theoretical soundness. Thus, having a notion of distance that is built on theoretically robust first principles and that is interpretable with respect to features ubiquitous in complex networks would allow for a meaningful comparison between different networks. Here we introduce a theoretically sound and efficient new measure of graph distance, based on the "length spectrum" function from algebraic topology, which compares the structure of two undirected, unweighted graphs by considering their non-backtracking cycles. We show how this distance relates to structural features such as presence of hubs and triangles through the behavior of the eigenvalues of the so-called non-backtracking matrix, and we showcase its ability to discriminate between networks in both real and synthetic data sets. By taking a topological interpretation of non-backtracking cycles, this work presents a novel application of Topological Data Analysis to the study of complex networks.

Motivation & Objective

• Address the lack of a canonical, theoretically grounded method for measuring structural dissimilarity between complex networks.
• Develop a graph distance measure that is interpretable in terms of key network features such as hubs and triangles.
• Ensure computational efficiency by relying only on the computation of a few largest eigenvalues of the non-backtracking matrix.
• Demonstrate the method's effectiveness in distinguishing real-world and synthetic networks across diverse data sets.
• Establish a novel link between topological data analysis—specifically the length spectrum—and complex network analysis.

Proposed method

• Define the length spectrum of a graph using the set of lengths of non-backtracking cycles, interpreted as free homotopy classes.
• Construct the non-backtracking matrix to encode the combinatorial structure of non-backtracking walks and compute its eigenvalues.
• Use the magnitude-ordered eigenvalues of the non-backtracking matrix as a proxy for the length spectrum, forming the basis of the TNBSD.
• Apply a truncation strategy to focus on the most significant eigenvalues, improving computational efficiency and robustness.
• Utilize complex eigenvalue distributions for visualization and further analysis, including Kernel PCA with cosine similarity to refine comparisons.
• Compare graphs by computing the L2 distance between their truncated eigenvalue sequences, enabling metric comparison.

Experimental results

Research questions

• RQ1 Can the length spectrum function, rooted in algebraic topology, serve as a principled foundation for graph distance in complex networks?
• RQ2 How do structural features like hubs and triangles influence the eigenvalue distribution of the non-backtracking matrix?
• RQ3 To what extent can the truncated eigenvalue spectrum of the non-backtracking matrix distinguish between real and synthetic network models?
• RQ4 Is the topological interpretation of non-backtracking cycles useful for enhancing interpretability and visualization of network distances?
• RQ5 How does TNBSD compare to existing graph distance measures in terms of discriminatory power, efficiency, and interpretability?

Key findings

• TNBSD effectively discriminates between real and synthetic networks, including in anomaly detection tasks, where it correctly identifies weeks with known events during the Enron scandal.
• The complex eigenvalue distributions of the non-backtracking matrix exhibit distinct geometric patterns across different network models (e.g., Erdős-Rényi, Barabási-Albert, Watts-Strogatz), enabling visual differentiation.
• The method is computationally efficient, requiring only the computation of a few largest eigenvalues of the non-backtracking matrix.
• TNBSD is interpretable: its behavior correlates with structural features such as the presence of hubs and triangles, which influence the eigenvalue spectrum.
• The method performs well in clustering and sampling tasks, with results showing that weekly aggregated Enron graphs are more similar to each other than to weekday graphs.
• Kernel PCA applied to eigenvalue projections improves comparison performance, suggesting that higher-order structure in the eigenvalue space enhances discriminatory power.