Framework for cascade size calculations on random networks.

Area of Science:

• Complex Systems and Network Science
• Statistical Physics
• Computational Social Science

Background:

• Cascade models are crucial for understanding phenomena like information diffusion and system failures.
• Existing methods, such as branching process approximations, have limitations in capturing complex cascade dynamics.
• Previous approaches struggled with networks exhibiting arbitrary degree distributions, correlations, and continuous or history-dependent cascade variables.

Purpose of the Study:

• To introduce a novel, exact framework for calculating cascade size evolution in random network ensembles.
• To extend the analysis beyond traditional branching process approximations.
• To accommodate complex network structures and cascade dynamics, including history-dependent and continuous quantities.

Main Methods:

• Developed an iterative probability distribution update method, shifting focus from branching processes.
• The framework is exact in the limit of infinite network size.
• Applies to network ensembles with arbitrary degree distributions, degree-degree correlations, and threshold distributions.

Main Results:

• The framework successfully calculates cascade size evolution for a broad class of cascade models.
• Demonstrated applicability to constant load models, which encompass many analytically tractable cascade models.
• Successfully analyzed a fiber bundle model, previously intractable with branching process approximations, revealing its full cascade dynamics.

Conclusions:

• The proposed framework offers an exact and versatile method for analyzing cascade dynamics in complex networks.
• It overcomes limitations of prior approximation techniques, enabling the study of more realistic and complex systems.
• The method's ability to capture historical dependencies and continuous quantities opens new avenues for research in network science and related fields.