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Homological percolation transitions in growing simplicial complexes
1CCSS, CTP, Department of Physics and Astronomy, Seoul National University, Seoul 08826, South Korea.
Chaos (Woodbury, N.Y.)
|July 12, 2021
Summary
This study explores homological percolation transitions in growing simplicial complexes, revealing how network growth and preferential attachment prevent topological localization. These findings offer topological insights into complex network maturation.
Area of Science:
- Complex Systems Science
- Network Theory
- Topology
Background:
- Simplicial complexes (SC) model higher-order interactions in complex systems like brain and social networks.
- Homological percolation transitions (HPTs) characterize the emergence of topological features (cycles and cavities) in these networks.
Purpose of the Study:
- To investigate HPTs in growing SCs using empirical data and a minimal model.
- To understand the topological dynamics and delocalization phenomena in maturing complex networks.
Main Methods:
- Exploration of HPTs using Betti numbers (first and second) to quantify cycles and cavities.
- Development of a minimal SC model incorporating growth and preferential attachment for social coauthorship networks.
- Analysis of delocalization by comparing simplex merging and birth rates.
Main Results:
- The minimal SC model successfully reproduces HPTs, identifying them as an infinite-order Berezinskii-Kosterlitz-Thouless type with distinct critical exponents.
- Unlike static SCs, growing SCs exhibit delocalization where the first Betti number increases post the appearance of the second Betti number.
- Delocalization is attributed to growth and preferential attachment, occurring when the rate of 2D simplex merging is less than the birth rate of isolated simplices.
Conclusions:
- Growing simplicial complexes display unique topological transitions and delocalization phenomena driven by network dynamics.
- The findings provide a topological perspective on the maturation processes in complex networks, including social and biological systems.
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