Mathematical analysis of topological and random m-order spread models

Area of Science:

• Complex Systems
• Mathematical Modeling

Background:

• Spreading processes are ubiquitous in nature and technology.
• Understanding the dynamics of these processes is crucial for prediction and control.
• Deterministic and random models offer different perspectives on propagation patterns.

Purpose of the Study:

• To analyze and compare two distinct models for spreading processes: one deterministic and one random.
• To investigate the spread rate of propagation patterns within each model.
• To explain observed differences in pattern frequency between the two models.

Main Methods:

• Utilizing substitution dynamical systems to model deterministic spreading.
• Employing branching processes to model random spreading.
• Encoding propagation patterns for quantitative analysis.

Main Results:

• Quantified the spread rate for various patterns in both deterministic and random models.
• Identified key factors contributing to the higher frequency of certain patterns in one model over the other.
• Demonstrated a significant difference in pattern prevalence between the deterministic and random approaches.

Conclusions:

• The choice of model (deterministic vs. random) significantly impacts the dynamics and observed frequencies of spreading patterns.
• Substitution dynamical systems and branching processes provide complementary frameworks for analyzing complex spreading phenomena.
• Numerical evidence supports the theoretical comparison of the two spreading models.