Networks Multiscale Entropy Analysis

TL;DR

This work addresses the limitation of single-scale entropy in capturing hierarchical patterns in complex networks by introducing a multiscale entropy framework built on spectral graph reduction. It combines spectral coarsening, lossless compression-based entropy estimates, and leave-one-out link-prediction analysis to quantify how structural entropy and predictability evolve across scales, using normalized baselines to enable cross-network comparisons. Key contributions include generalizing compression-based entropy to multiscale representations, identifying domain-specific entropy regimes (stable, increasing, hybrid), and demonstrating that multiscale entropy substantially improves the prediction of link-prediction scores (e.g., $R^2$ up to $0.92$). The approach is shown to be scalable to large networks and provides a principled tool for network characterization, classification, and inference, with potential extensions to directed/temporal networks and alternative multiscale descriptors. Overall, multiscale entropy links compressibility and predictability across hierarchical network representations, offering new insights into the informational geometry of complex systems.

Abstract

Understanding the structural complexity and predictability of complex networks is a central challenge in network science. Although recent studies have revealed a relationship between compression-based entropy and link prediction performance, existing methods focus on single-scale representations. This approach often overlooks the rich hierarchical patterns that can exist in real-world networks. In this study, we introduce a multiscale entropy framework that extends previous entropy-based approaches by applying spectral graph reduction. This allows us to quantify how structural entropy evolves as the network is gradually coarsened, capturing complexity across multiple scales. We apply our framework to real-world networks across biological, economic, social, technological, and transportation domains. The results uncover consistent entropy profiles across network families, revealing three structural regimes$\unicode{x2013}$stable, increasing, and hybrid$\unicode{x2013}$that align with domain-specific behaviors. Compared to single-scale models, multiscale entropy significantly improves our ability to determine network predictability. This shows that considering structural information across scales provides a more complete characterization of network complexity. Together, these results position multiscale entropy as a powerful and scalable tool for characterizing, classifying, and assessing the structure of complex networks.

Figures (12)

• Figure 1: Multiscale entropy for different network families. The graphs are of 2500 nodes.
• Figure 2: Entropy trajectories across real-world network families stratified by node count. Each plot illustrates the evolution of normalized length compression entropy under successive reductions (80%, 60%, 40%, 20%) for networks in a specific domain.
• Figure 3: Entropy evolution across transport, technological, and informational networks, stratified by network size.
• Figure 4: Visualization of clusters identified by K-means (k=3) using PCA projection of the five-dimensional entropy vectors. Each point represents a graph and is colored by cluster assignment.
• Figure 5: Multiscale Entropy