[Paper Review] From random walks to distances on unweighted graphs

Tatsunori Hashimoto, Yi Sun+1 more|arXiv (Cornell University)|Nov 2, 2015

Complex Network Analysis Techniques21 references6 citations

TL;DR

This paper introduces the Laplace transformed hitting time (LTHT) as a robust, consistent metric for measuring similarity on unweighted directed graphs, using stochastic calculus to establish a correspondence between graph random walks and Brownian motion. The LTHT avoids the degeneracy of expected hitting times, preserves clustering, and remains robust under non-geometric edge perturbations, outperforming alternatives in both theory and simulation.

ABSTRACT

Large unweighted directed graphs are commonly used to capture relations between entities. A fundamental problem in the analysis of such networks is to properly define the similarity or dissimilarity between any two vertices. Despite the significance of this problem, statistical characterization of the proposed metrics has been limited. We introduce and develop a class of techniques for analyzing random walks on graphs using stochastic calculus. Using these techniques we generalize results on the degeneracy of hitting times and analyze a metric based on the Laplace transformed hitting time (LTHT). The metric serves as a natural, provably well-behaved alternative to the expected hitting time. We establish a general correspondence between hitting times of the Brownian motion and analogous hitting times on the graph. We show that the LTHT is consistent with respect to the underlying metric of a geometric graph, preserves clustering tendency, and remains robust against random addition of non-geometric edges. Tests on simulated and real-world data show that the LTHT matches theoretical predictions and outperforms alternatives.

Motivation & Objective

• To address the lack of statistical consistency in existing random walk-based graph similarity metrics, particularly the degeneracy of expected hitting time.
• To develop a theoretically grounded, well-behaved alternative to expected hitting time that preserves geometric and clustering structure in graphs.
• To establish consistency, cluster preservation, and robustness of the LTHT under a general latent metric space model for graphs.
• To provide rigorous theoretical justification for quasi-walk metrics using stochastic processes and continuum limits.
• To demonstrate that the LTHT converges to shortest path distance or consistent path-averaged metrics under appropriate scaling.

Proposed method

• Develops a correspondence between hitting time functionals on graphs and limiting Itô processes, enabling stochastic calculus analysis.
• Constructs a weighted random walk on graphs whose limit is a Brownian motion, facilitating continuous-time approximation.
• Applies Laplace transformation to hitting times to define the LTHT, which stabilizes degenerate behavior of expected hitting times.
• Uses continuum limit analysis as n → ∞ and εn → 0, with scaling constants gn and deterministic ε(x) to model local neighborhood structure.
• Employs moment bounds and concentration inequalities (e.g., Chebyshev) to control fluctuations in neighborhood size and edge counts.
• Analyzes the RA index robustness via decomposition of neighbor counts into geometric and noise components, with bounds on Cij under different distance regimes.

Experimental results

Research questions

• RQ1 Can a hitting-time-based metric be consistently defined on unweighted graphs without degeneracy?
• RQ2 Does the LTHT converge to the shortest path distance or a consistent path-averaged metric in the scaling limit?
• RQ3 How does the LTHT preserve clustering structure in geometric graphs under perturbations?
• RQ4 Is the LTHT robust to random non-geometric edges when the majority of edges reflect latent geometry?
• RQ5 Can the RA index be shown to reliably recover latent distances under the same model assumptions?

Key findings

• The LTHT avoids the degeneracy of expected hitting time by using a Laplace transform, which stabilizes the metric under scaling limits.
• The LTHT converges to the shortest path distance in the limit as the parameter β is scaled appropriately, as shown in Theorem S5.2.
• The LTHT is a consistent estimator of the underlying latent metric distance under the geometric graph model, as proven in Theorem 4.5.
• The LTHT preserves clustering tendency, as formalized in Theorem 4.6, by respecting local community structure in the graph.
• The LTHT remains robust to random non-geometric edges, with Theorem 4.9 showing it still recovers similarity queries accurately.
• The RA index is robust: for |xi − xj| < min{εn(xi), εn(xj)}, hnRij → cij > 0, while for |xi − xj| > 2 max{εn(xi), εn(xj)}, hnRij → 0, confirming its ability to distinguish close vs. distant pairs.