A Model of Synchronization for Self-Organized Crow
The paper proposes a unified, non‑probabilistic framework for describing synchronization in self‑organized crowds—systems of autonomous agents without a central authority—by modeling a feedback loop among three dynamic quantities: individual agent action (S), macro‑level observation (O), and external force (E). The interaction is mediated by three response coefficients (A, B, C) that may be linear or nonlinear. Agents are assumed to switch between a low‑coupling "normal" state and a high‑coupling "reactive" state, with the proportion of reactive agents (NH/N) governing the aggregate coupling coefficient B. An order parameter R (and an equivalent correlation‑based measure ρc) quantifies the degree of crowd synchronization, shown analytically to increase with NH/N and to decline with random noise. A critical feedback product AB=1 marks a tipping point where the crowd becomes self‑amplifying and unstable, leading to phenomena such as financial bubbles, market momentum, bridge sway, and animal herding. The model is applied to simulate bubble formation, trendiness (Td), and volatility, demonstrating how external forces, agent state transitions, and the AB product drive exponential growth or decay of observations. The paper discusses practical implications for monitoring market fragility via the reactive‑agent fraction and AB_max, links to existing models (Vicsek, Ising, ARCH/GARCH), and suggests future work on data‑driven calibration and policy interventions.
Topics
Feedback Loop Formalism Linking Agent Action, Observation, and External Force
Defines the core model with variables S (agent action), O (macro observation), E (external force) and response coefficients A, B, C, including linear and nonlinear formulations and the governing differential equations.
Two‑State Agent Coupling: Normal vs Reactive Dynamics
Introduces the hypothesis that agents switch between low‑coupling (B_L) and high‑coupling (B_H) states, derives B as a linear function of the reactive‑agent proportion N_H/N, and discusses psychological and biological motivations.
Synchronization Metrics: Order Parameter R and Correlation Measure ρ_c
Presents the order parameter R (ratio of aligned actions) and an equivalent crowd‑level correlation ρ_c, derives their dependence on N_H/N and noise, and connects them to weighted sums of the agents’ volatility and correlation matrix.
Critical Tipping Point (AB=1) and Self‑Amplifying Instability
Shows that when the product of feedback coefficients AB approaches unity the system becomes hyper‑sensitive, defines the critical reactive‑agent fraction N_HC, and links this to bubble formation and phase transitions.
Simulation of Financial Bubbles, Trendiness, and Volatility
Describes discrete‑time simulations illustrating scenarios with AB<1, AB>1, and AB=1, demonstrating how external forces, reactive‑agent dynamics, and feedback produce exponential price growth, crashes, and varying trend (T_d) and volatility (σ_c).
Cross‑Disciplinary Connections to Vicsek, Ising, and ARCH/GARCH Models
Compares the proposed framework to established models in physics (Vicsek, Ising), economics (ARCH/GARCH), and agent‑based simulations, highlighting similarities and extensions.