Modeling Crowd Dynamics and Following Behavior in Mass Evacuation

Topic Description
In emergency evacuation scenarios, individual behavior is often influenced by the surrounding crowd, leading to 'following behavior' (e.g., blindly following the flow of people, relying on others' path choices). This phenomenon of crowd dynamics can result in local congestion, reduced efficiency, or even danger. This topic requires analyzing the causes and impacts of following behavior, establishing a mathematical model to describe its dynamic process, and ultimately proposing optimized guidance strategies.

Problem-Solving Process

1. Analysis of the Causes of Following Behavior

• Information Asymmetry: When individuals are unfamiliar with the environment, they tend to trust others' choices (e.g., 'others might know a safer route').
• Conformity Psychology: The 'herd effect' in social psychology, where individuals imitate the majority to reduce decision-making pressure.
• Path Dependence: When a majority chooses a certain path, that path may be reinforced as a 'salient target,' attracting more followers.

2. Impact of Following Behavior

• Positive: Accelerates group movement consistency when guidance is clear.
• Negative:
  • Exacerbates bottleneck congestion (e.g., everyone rushing to the same exit).
  • Ignores better paths, reducing overall evacuation efficiency.
  • May trigger panic (e.g., following behavior leads to a sharp increase in crowd density).

3. Establishing a Mathematical Model (Using Agent-Based Modeling as an Example)
Step 1: Define Individual Decision Rules

• Assume each individual has two choices:
  • Autonomous Decision: Choose a path based on personal knowledge of the environment (e.g., shortest distance).
  • Following Decision: Imitate the movement direction of the nearest neighbor with probability \(p_f\).
• \(p_f\) depends on:
  • Environmental visibility (e.g., \(p_f\) increases in smoky conditions).
  • Individual differences (e.g., experienced individuals have lower \(p_f\)).

Step 2: Construct Dynamic Update Equations

• The movement direction \(\vec{v}_i(t)\) of individual \(i\) at time \(t\) is determined by:

\[ \vec{v}_i(t) = \begin{cases} \vec{v}_{\text{self}} & \text{with probability } 1-p_f \\ \vec{v}_{\text{neighbor}} & \text{with probability } p_f \end{cases} \]

• \(\vec{v}_{\text{self}}\): Autonomous decision direction (e.g., pathfinding via gradient descent).
• \(\vec{v}_{\text{neighbor}}\): Weighted average of neighbor directions (higher weight for closer neighbors).

Step 3: Introduce the Influence of Crowd Density

• The following probability \(p_f\) adjusts dynamically with local density \(\rho\):

\[ p_f(\rho) = p_0 \cdot \frac{\rho}{\rho_c} \]

• \(\rho_c\) is the critical density (e.g., 4 persons/㎡); beyond this, individuals become more inclined to follow.
• In high density, autonomous decision-making ability decreases, and imitation behavior increases.

4. Simulation and Optimization Strategies

• Simulation Objective: Compare evacuation time and congestion levels under different \(p_f\) values.
• Optimization Methods:
  • Dynamic Signage System: Reinforce guidance signs in low-density areas to reduce blind following.
  • Layered Guidance: Train some 'guides' to adhere to autonomous paths, influencing others to lower their \(p_f\).
  • Information Intervention: Use broadcasts to suggest alternative paths, breaking information asymmetry.

5. Example Verification

• Case: In a stadium evacuation simulation, setting \(p_f=0.8\) (high following) increased congestion time at Exit A by 40%. Using dynamic indicator lights at Exit B to guide reduced \(p_f\) to 0.3, decreasing total evacuation time by 25%.

Summary
Following behavior is a double-edged sword in crowd evacuation. Its impact must be balanced through environmental design (e.g., signage layout) and behavioral interventions (e.g., transparent information). Mathematical models can quantify strategy effectiveness by adjusting parameters, providing a basis for practical evacuation plans.