Crossover behavior in driven cascades.

Area of Science:

• Complex Systems Science
• Statistical Physics
• Network Science

Background:

• Cascading failures are prevalent in various systems, from ecosystems to financial markets.
• Understanding the statistical distributions of cascade sizes is crucial for risk assessment.
• Previous models often assume static system properties, limiting their applicability to dynamic scenarios.

Purpose of the Study:

• To develop a theoretical model explaining power-law crossover behavior in systems prone to cascading failure.
• To introduce and define the concept of "propagation power" as a key determinant of cascade dynamics.
• To analytically derive the exponents governing cascade size distributions before and after crossover.

Main Methods:

• Modeling cascading failure using a continuous state branching process.
• Introducing "propagation power" as a dynamic variable following an upward drifting Brownian motion with discontinuous drops.
• Analyzing the cascade size distribution by averaging over the distribution of propagation power.

Main Results:

• Demonstrated that power-law crossover behavior naturally emerges from the interplay of dynamic propagation power and cascading events.
• Identified a critical level of propagation power where pure power-law behavior is observed and mean cascade size diverges.
• Showed that system-wide averaging over propagation power distributions leads to observable crossover in cascade size distributions.

Conclusions:

• The proposed model provides a unified framework for understanding power-law crossover in cascading failure systems.
• The concept of propagation power offers a quantifiable metric for system susceptibility and cascade dynamics.
• The analytical results offer insights into the statistical properties of complex systems and their failure modes.