Spatial Topology and Network Flow Optimization in Crowd Evacuation

Spatial Topology and Network Flow Optimization in Crowd Evacuation

Problem Description
During evacuation in large public spaces (such as stadiums, subway stations), people need to move through a network formed by spatial nodes like corridors, staircases, and exits. Spatial topology refers to the connection relationships between these nodes (e.g., path length, width, number of turns), while network flow optimization aims to allocate pedestrian flow using mathematical methods to minimize total evacuation time or avoid local congestion. This problem requires analyzing how topology affects evacuation efficiency and designing network flow optimization strategies.

Solution Process

1. Understanding Basic Elements of Spatial Topology

  • Nodes: Represent key locations in space (e.g., room centers, staircase entrances, exits).
  • Edges: Paths connecting nodes (e.g., hallways, passageways), with attributes (length, width, travel speed).
  • Topological Features:
    • Connectivity: Whether paths exist between nodes (e.g., dead ends reduce evacuation efficiency).
    • Redundancy: The number of alternative paths (high redundancy can分流 congestion).
    • Bottlenecks: Nodes with sudden width reductions or excessive turns (prone to forming congestion points).

Example: In a stadium's topology, paths from stands to exits may include multiple staircases (nodes) and passageways (edges), where the width of exit corridors is a key bottleneck.

2. Establishing a Network Flow Model

  • Abstract the evacuation problem as a directed graph:
    • Node capacity: Determined by the maximum number of people the physical space can hold.
    • Edge capacity: Calculated based on corridor width and travel speed (e.g., maximum number of people passing per unit time).
  • Objective Function: Minimize the maximum flow time (i.e., the shortest time for all people to reach safe areas).
  • Constraints:
    • Flow conservation (number of people entering a node equals those leaving).
    • Edge capacity limits (flow does not exceed the maximum passage capacity of the corridor).

Formula Example:
Let the node set be \(V\), edge set be \(E\), \(c(e)\) be the capacity of edge \(e\), and \(f(e)\) be the actual flow. The optimization goal is:

\[\min \max_{v \in V} T(v) \]

where \(T(v)\) is the time for people to reach node \(v\).

3. Analyzing the Impact of Topology

  • Critical Path Identification: Use shortest path algorithms (e.g., Dijkstra) to find the longest path from all people to the exit, which determines the theoretical minimum evacuation time.
  • Bottleneck Analysis: Identify congestion points by calculating edge "utilization" (flow/capacity). For example, if a passage's utilization consistently approaches 1, alternative paths or capacity expansion are needed.
  • Topology Improvement:
    • Add shortcuts: Connect distant areas directly to exits.
    • Reduce turns: Turns slow travel speed and can be mitigated by optimizing corridor shapes.

4. Network Flow Optimization Methods

  • Maximum Flow Algorithm (e.g., Ford-Fulkerson): Used to calculate the theoretical maximum passage capacity of the network but requires adjustment based on personnel distribution.
  • Dynamic Flow Allocation:
    • Phased Scheduling: Guide people in batches based on their distance to the exit to avoid simultaneous crowding at bottlenecks.
    • Real-time Rerouting: Dynamically assign paths based on congestion (e.g., using electronic signs to guide detours).
  • Heuristic Strategies:
    • Capacity-Weighted Path Allocation: Prioritize assigning people to paths with high capacity and low congestion.
    • Reverse Flow Control: Temporarily restrict inflow at certain entrances to prevent逆向冲突.

5. Simulation Verification and Parameter Adjustment

  • Use agent-based simulation software (e.g., AnyLogic) to simulate the effects of different topologies and flow strategies.
  • Sensitive Parameters: People's movement speed, node capacity, path selection preferences (e.g., tendency to choose shortest paths).
  • Optimization Iteration: Adjust topology (e.g., widen bottleneck corridors) or flow allocation rules based on simulation results.

Example Application
In subway station evacuation, topology analysis identifies staircases between platforms and exits as bottlenecks. Optimization solutions include:

  1. Adding temporary handrails to separate ascending and descending crowds, improving staircase efficiency.
  2. Using waiting areas as缓冲 zones to release people in batches.
  3. Dynamically adjusting gate directions to make some exits exit-only.

Summary
Spatial topology determines the physical constraints of the evacuation network, while network flow optimization enhances actual efficiency through mathematical modeling and dynamic strategies. Both must be combined with simulation tools for反复验证 to design robust evacuation plans.