[Paper Review] Fast computation of von Neumann entropy for large-scale graphs via quadratic approximations

Hayoung Choi, Jinglian He+2 more|arXiv (Cornell University)|Nov 12, 2018

Complex Network Analysis Techniques26 references4 citations

TL;DR

This paper proposes novel quadratic approximations to accelerate von Neumann graph entropy (VNGE) computation in large-scale graphs, reducing the time complexity from O(n³) to O(n + m). By approximating the entropy function −λᵢ ln λᵢ using quadratic forms, the method enables linear-time computation without requiring the largest eigenvalue, outperforming prior approaches like FINGER in accuracy and efficiency across random and real-world networks.

ABSTRACT

The von Neumann graph entropy (VNGE) can be used as a measure of graph complexity, which can be the measure of information divergence and distance between graphs. However, computing VNGE is extensively demanding for a large-scale graph. We propose novel quadratic approximations for fast computing VNGE. Various inequalities for error between the quadratic approximations and the exact VNGE are found. Our methods reduce the cubic complexity of VNGE to linear complexity. Computational simulations on random graph models and various real network datasets demonstrate superior performance.

Motivation & Objective

• To address the high computational cost of exact von Neumann graph entropy (VNGE) computation in large-scale graphs, which currently scales as O(n³).
• To develop efficient, model-free approximations of VNGE that avoid expensive eigenspectrum computation.
• To achieve linear-time complexity O(n + m) by leveraging quadratic approximations of the entropy function −λᵢ ln λᵢ.
• To evaluate the accuracy and efficiency of the proposed approximations against exact VNGE on synthetic and real-world graph datasets.
• To demonstrate the practical utility of fast VNGE computation in applications such as Jensen-Shannon distance for graph similarity.

Proposed method

• Propose four quadratic approximations: Modified Taylor, Radial Projection, FINGER-inspired, and Mixed Quadratic, to approximate the nonlinearity −λᵢ ln λᵢ in VNGE computation.
• Use the purity of the density matrix, computable in O(n + m) time, to evaluate the approximations without requiring full eigenspectrum computation.
• Derive theoretical error bounds for the Modified Taylor and Radial Projection approximations, establishing their reliability.
• Apply the approximations to estimate VNGE using only the trace and Frobenius norm of the normalized Laplacian matrix, avoiding eigenvalue decomposition.
• Use Radial Projection to avoid dependency on the maximum eigenvalue, improving efficiency for sparse graphs.
• Integrate the approximations into the Jensen-Shannon distance framework for graph similarity, enabling scalable comparison of large graph sequences.

Experimental results

Research questions

• RQ1 Can quadratic approximations of the entropy function −λᵢ ln λᵢ be used to compute von Neumann graph entropy in linear time O(n + m)?
• RQ2 How do the proposed approximations compare in accuracy and efficiency to existing methods like FINGER, especially in terms of error bounds and eigenvalue dependency?
• RQ3 Does the absence of maximum eigenvalue computation in Radial Projection lead to superior performance on sparse real-world graphs?
• RQ4 What is the correlation between exact VNGE and the approximations across diverse random graph models (ER, BA, WS)?
• RQ5 Can the fast VNGE computation enable scalable graph similarity and anomaly detection using Jensen-Shannon distance?

Key findings

• The Mixed Quadratic approximation achieves the highest correlation with exact VNGE across all tested random graph models, outperforming FINGER and other approximations.
• Radial Projection shows strong performance without requiring the maximum eigenvalue, achieving high accuracy and linear-time complexity on sparse graphs.
• On real-world unweighted and weighted networks, the Mixed Quadratic and Radial Projection approximations exhibit superior accuracy compared to FINGER and Modified Taylor.
• The computational time of the proposed methods scales linearly with graph size, reducing O(n³) complexity to O(n + m), enabling practical use on large-scale graphs.
• The Jensen-Shannon distance computed using the proposed approximations maintains high fidelity to the exact version, supporting its use in anomaly and bifurcation detection.
• Theoretical error bounds are established for Modified Taylor and Radial Projection approximations, providing confidence in their reliability for large-scale applications.