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Fractal networks: Topology, dimension, and complexity
1School of Mathematics, Georgia Institute of Technology, 686 Cherry St NW, Atlanta, Georgia 30332, USA.
Chaos (Woodbury, N.Y.)
|April 10, 2024
Summary
This study introduces a graph theory framework to rigorously analyze fractal networks. It defines fractal dimensions using combinatorial methods, linking network self-similarity to community intersection patterns.
Area of Science:
- Network Science
- Graph Theory
- Fractal Geometry
Background:
- Growing interest in self-similarity and fractality in complex networks.
- Challenges in applying fractal geometry principles to finite, discrete structures.
- Need for mathematically rigorous methods to analyze fractal networks.
Purpose of the Study:
- To present a graph-theoretical framework for identifying and analyzing fractal networks.
- To establish direct graph-theoretical analogs of topological and fractal dimensions.
- To bridge discrete and continuous definitions of fractal dimensions.
Main Methods:
- Utilizing graph theory and combinatorics to analyze network structures.
- Defining graph-theoretical analogs of Lebesgue and Hausdorff dimensions.
- Connecting fractal properties to combinatorial parameters like graph colorings.
Main Results:
- Demonstrated that network self-similarity arises from densely connected community intersections.
- Established a combinatorial characterization of Lebesgue dimension for graph representations.
- Provided a rigorous definition of fractal networks within a combinatorial context.
Conclusions:
- The developed theory offers a robust method for analyzing fractal characteristics in complex networks.
- This framework facilitates the application of combinatorial methods to real-world network analysis.
- It lays the groundwork for future research into fractal properties of networks.
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