Network Structure & Utilization Educational Workspace

These matrices represent the physical topology of the network:

• A (Weighted Adjacency Matrix): Shows the direct connectivity between nodes.
• P (Path Matrix): Shows both the direct and indirect connectivity between nodes across the entire network.
• E (External Matrix): Shows strictly the indirect connectivity between nodes (E = P - A).
• Binarized Operations: Converts weighted capacities into a binary format (1 for connection, 0 for none).

These matrices represent how trajectories (e.g., commuter routes) utilize the physical topology. The system separates routes based on the state of the Adjacency Matrix ($A$):

• F (Flow Matrix): Only direct link flow.
• L (Indirect Flow Matrix): Only indirect link flowIndirect link flow only. The total indirect flow is the sum of Alternative and Substitute routes ($L = T + S$).
• D (Generalized Origin-Destination Matrix): Both direct and indirect link flow.
• T (Alternative Route Matrix): Flow utilizing indirect links while a direct link exists.
• S (Substitute Route Matrix): Flow utilizing indirect links because the direct link does not exist.

The Indirect Flow Split:

• T (Alternative Route Matrix): Occurs when a trajectory uses an indirect path, but a direct physical link actually exists ($A_{ij} > 0$). This often represents commuters bypassing a direct highway to take backroads due to congestion.
• S (Substitute Route Matrix): Occurs when a trajectory uses an indirect path because no direct physical link exists ($A_{ij} = 0$). The commuter is forced to use an intermediate node.

Urban Planning Scenario: In real life, a trajectory is an ordered sequence of nodes representing a commuter's path (e.g., Suburb a → Highway b → Downtown c).

Challenge: Navigate to the Network Utilization tab. Inject a trajectory that forces traffic through a single central node. Observe the Utilization Matrices to see which route becomes an "Alternative" (T) or "Substitute" (S) route. Then, move to the Relationships tab and execute the proofs to see how A, P, F, L, and D mathematically interact!

You do not need a degree in discrete mathematics to use this tool. Think of the network as a plumbing system:

• Matrix A (Adjacency): The physical pipes you have installed. If $A_{ij} = 1$, a pipe connects tank $i$ directly to tank $j$.
• Matrix P (Path): Can water eventually get from tank $i$ to tank $j$, even if it has to pass through 10 other tanks first? If yes, $P_{ij} \geq 1$.
• Matrix D (Generalized OD): Where the water actually needs to go, regardless of how the pipes are laid out.
• Formula Verification ($F + T = A \cdot D$): This simply proves that the total traffic utilizing direct roads ($F$) plus the traffic forced to use alternative routes ($T$) must mathematically equal the physical capacity ($A$) intersecting with human demand ($D$).