A Matrix Formalism for Competing Affiliations

Area of Science:

• Sociophysics
• Mathematical Modeling
• Population Dynamics

Background:

• Competition between social affiliations is common in human societies.
• Understanding the dynamics of adherence and switching is crucial for social sciences.

Purpose of the Study:

• To develop a mathematical framework for analyzing competition between two social affiliations.
• To characterize the conditions for different population dynamics (dominance, coexistence, bistability).
• To analyze the impact of altering affiliation fitness on these dynamics.

Main Methods:

• A compact matrix formalism was developed.
• A one-dimensional replicator equation driven by a 2x2 interaction matrix was used.
• Regime-switch analysis of rank-one matrix alteration was performed.

Main Results:

• Two scalar parameters fully characterize the qualitative regimes of affiliation competition.
• A single control quantity determines the effect of fitness alterations on regime parameters.
• Threshold conditions for qualitative changes in dynamics were identified.
• A persistent loyal core of adherents prevents extinction and reduces intervention needed for recovery.

Conclusions:

• The developed matrix formalism provides a robust tool for analyzing social affiliation dynamics.
• Fitness alterations can predictably switch population regimes.
• The size of a loyal core and alteration strength have an inverse relationship in recovery dynamics.