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Global topological synchronization of weighted simplicial complexes.
Runyue Wang1, Riccardo Muolo2, Timoteo Carletti3
1School of Mathematical Sciences, <a href="https://ror.org/026zzn846">Queen Mary University of London</a>, London E1 4NS, United Kingdom.
Physical Review. E
|August 20, 2024
Summary
Weighted simplicial complexes enable global synchronization of topological signals, even on odd-dimensional structures like edges. This finding advances understanding of collective phenomena in complex systems.
Area of Science:
- Complex Systems Science
- Network Theory
- Topology
Background:
- Higher-order networks, such as simplicial complexes, model many-body interactions and complex system dynamics.
- Simplicial complexes support topological signals on nodes, edges, and higher-dimensional elements.
- Global synchronization of topological signals is crucial but limited by network topology.
Purpose of the Study:
- Investigate global topological synchronization on weighted simplicial complexes.
- Determine if weights overcome limitations in unweighted higher-order networks.
- Explore synchronization on odd-dimensional structures like edge signals.
Main Methods:
- Analysis of weighted simplicial complexes, including the weighted triangulated torus and waffle.
- Characterization of higher-order spectral properties.
- Geometric interpretation of weights and synchronization phenomena.
Main Results:
- Topological signals can globally synchronize on weighted simplicial complexes, overcoming limitations of unweighted cases.
- Synchronization is achievable even for odd-dimensional signals (e.g., edge signals).
- Specific weighted complexes (torus, waffle) are shown to sustain global edge signal synchronization under certain weight conditions.
Conclusions:
- Weighted simplicial complexes offer enhanced capabilities for observing collective phenomena like synchronization compared to unweighted ones.
- Weights can be geometrically interpreted, linking them to properties like edge lengths in curved simplices.
- The study expands the understanding of synchronization dynamics in higher-order network structures.
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