A Novel Framework for Multiscale Fractal Analysis in Complex Networks

Thesis Abstract

The complexity inherent in modern networks, such as social, biological, and technological systems, necessitates advanced analytical frameworks to capture their multiscale structural intricacies and dynamic behaviors. Traditional tools often fall short in accurately characterizing the fractal geometries and hierarchical features present at different scales within these networks. This study addresses this challenge by developing a novel multiscale fractal analysis framework, aimed at providing a comprehensive and scalable method for measuring the fractal dimensions and self-similarity patterns across multiple levels of complex networks. The primary objective is to formulate and validate an analytical model that integrates multifractal spectrum analysis with hierarchical network decomposition techniques, thereby enhancing the understanding of multiscale structures in complex systems. The research adopts a mixed-methods approach, combining quantitative fractal analysis with qualitative evaluation of network topologies. The quantitative component involves data collection from a diverse sample of 150 complex networks, including social interaction networks, protein-protein interaction networks, and transport systems, obtained from publicly available repositories such as the Stanford Network Analysis Project, the BioGRID database, and regional transportation authorities. The sample size was determined to ensure representativeness across different network types and scales, with stratified random sampling used to select datasets exhibiting a range of connectivity and density features. Data collection instruments include network data matrices and metadata files, which are processed through custom-developed algorithms embedded in Python and MATLAB environments. The analysis employs advanced techniques such as the box-counting method for initial fractal dimension estimation, combined with a hierarchical clustering algorithm to decompose networks into multiscale modules. Multifractal spectrum analysis is performed to identify the heterogeneity and complexity of the network structures. The framework also incorporates the use of stability analysis to assess the robustness of fractal measures across different scales. Statistical tests, including repeated measures ANOVA and regression analysis, are used to evaluate the significance of variability and identify factors influencing fractal characteristics. The model specification is based on theoretical underpinnings from the Self-Organized Criticality (SOC) theory and the Renormalization Group (RG) approach, which support the hierarchical nature of fractal phenomena within networks. Expected findings include the identification of consistent multiscale fractal signatures within different types of complex networks, revealing that these signatures are associated with network functionality, resilience, and resilience. It is anticipated that the proposed framework will demonstrate superior accuracy and scalability in capturing multiscale fractal features compared to existing single-scale methods. The study aims to contribute novel insights into the spatial and hierarchical organization of complex systems, advancing methodological approaches in network science and fractal geometry. Additionally, the integration of multifractal analysis with hierarchical decomposition is expected to establish new paradigms for multiscale network modeling, with implications for diagnostics, monitoring, and optimization of complex systems. The main conclusion underscores the efficacy of the proposed multiscale fractal framework in uncovering hidden structural complexities across diverse network systems. Recommendations include the incorporation of the framework into existing network analysis tools for improved system diagnostics and the extension of the approach to dynamic network analysis with temporal fractal measures. Future research directions involve applying the framework to real-time data streams, exploring its potential in predictive modeling, and further refining the theoretical links between fractal properties and system behavior. Overall, the study provides a significant advancement in the analytical toolkit available for understanding the multiscale nature of complex networks and offers a foundation for subsequent empirical and theoretical explorations in this evolving field.