Forward reachable sets: Analytically derived properties of connected components for dynamic networks.
Benjamin Armbruster1, L I Wang2, Martina Morris2
1Northwestern University, Industrial Engineering and Management Sciences, Evanston, IL, USA.
Network Science (Cambridge University Press)
|September 5, 2022
Summary
We analyzed forward reachable sets (FRS) in dynamic networks, finding their growth depends on network structure and edge duration. This work offers insights into epidemic modeling and network dynamics.
Area of Science:
- Network Science
- Complex Systems
- Mathematical Biology
Background:
- Formal analysis of emergent structural properties in dynamic networks remains an underdeveloped area.
- Forward reachable sets (FRS) are a key measure for understanding information propagation and connectivity in dynamic networks.
- Existing methods often do not account for temporal dynamics or detailed network structures.
Purpose of the Study:
- To formally analyze the emergent structural properties of dynamic networks using forward reachable sets (FRS).
- To investigate the influence of degree distribution and edge duration on FRS growth in temporal networks.
- To provide a theoretical framework applicable to epidemic modeling and other dynamic network processes.
Main Methods:
- Developed a stochastic framework to derive closed-form expressions for the mean and variance of FRS exponential growth rates.
- Incorporated both edge and node dynamics within the temporal network model.
- Utilized simulation and approximation to explore finite population effects and variations across degree distributions (Poisson, Bernoulli).
Main Results:
- Derived closed-form expressions for the growth rate of FRS in temporal networks with edge and node dynamics.
- Identified thresholds for FRS growth in networks exhibiting node dynamics.
- Demonstrated how FRS properties are influenced by edge duration and cross-sectional degree distributions.
Conclusions:
- The study provides a robust theoretical foundation for analyzing FRS in dynamic networks.
- Findings offer a direct link to epidemic modeling, where FRS size serves as an upper bound for epidemic spread.
- The derived mathematical framework enhances understanding of network structure-dynamics interplay.
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