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Early warning signals for percolation transitions in networks
A V Goltsev1, S N Dorogovtsev1
1University of Aveiro, Department of Physics, I3N, 3810-193 Aveiro, Portugal and Ioffe Physical-Technical Institute, 194021 St. Petersburg, Russia.
Physical Review. E
|June 19, 2026
Summary
This study introduces a new method using "observers" to measure network susceptibility, providing an early warning signal for percolation phase transitions in complex networks. This approach helps predict critical points despite finite-size effects.
Area of Science:
- Network Science
- Statistical Physics
- Complex Systems
Background:
- Percolation phase transitions are crucial in complex networks but difficult to predict due to finite-size effects.
- Identifying the critical point of giant connected component appearance/disappearance is challenging.
Purpose of the Study:
- To develop a novel method for computing and measuring network susceptibility.
- To provide an early warning signal for approaching phase transitions in complex networks.
Main Methods:
- Introduction of "observers" (randomly chosen nodes) to monitor local network connectivity.
- Derivation of explicit equations for susceptibility calculation.
- Analysis of critical behavior near phase transitions in various network types.
Main Results:
- A strong increase in susceptibility serves as an early warning signal for phase transitions.
- The method is effective for uncorrelated random networks, networks with dependency links, and k-cores.
- Demonstrated susceptibility calculation and critical behavior analysis.
Conclusions:
- The developed susceptibility measurement method reliably predicts percolation phase transitions.
- The universality of critical behavior is supported by phenomenological Landau theory.
- This method enhances understanding and prediction of network dynamics.
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