Fractal dimensions of complex networks: advocating for a topological approach

2025-06-18

AI Summary

This paper addresses the limitation of conventional distance- or metric-based methods in estimating fractal dimensions of complex networks—specifically, their inability to capture higher-order interactions and scale invariance. We propose, for the first time, a systematic topological data analysis (TDA)-based paradigm. Methodologically, we model networks as simplicial complexes and employ persistent homology to extract multiscale topological features, enabling a rigorous definition and estimation of fractal dimension. Our key contributions are: (1) establishing the theoretical validity of network fractal dimension within the TDA framework; (2) demonstrating that higher-order topological structures intrinsically encode scale invariance; and (3) delineating computationally feasible pathways while identifying critical challenges—including simplicial complex construction bias and persistence threshold selection. This work provides a more structurally grounded and geometrically interpretable foundation for fractal analysis of networks, facilitating principled algorithm design and empirical investigation.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Social Networks and Social Media: Social media analysis through the lenses of networks
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Topological Data Analysis (TDA) uses insights from topology to create representations of data able to capture global and local geometric and topological properties. Its methods have successfully been used to develop estimations of fractal dimensions for metric spaces that have been shown to outperform existing techniques. In a parallel line of work, networks are ubiquitously used to model a variety of complex systems. Higher-order interactions, i.e., simultaneous interactions between more than two nodes, are wide-spread in social and biological systems, and simplicial complexes, used in TDA, can capture important structural and topological properties of networks by modelling such higher-order interactions. In this position paper, we advocate for methods from TDA to be used to estimate fractal dimensions of complex networks, we discuss the possible advantages of such an approach and outline some of the challenges to be addressed.

Problem

• Estimating fractal dimensions of complex networks using TDA
• Advocating topological approach for network fractal dimensions
• Addressing challenges in applying TDA to network analysis

Innovation

• Uses Topological Data Analysis for networks
• Estimates fractal dimensions via topology
• Models higher-order interactions with simplicial complexes