Structural controllability
Technical note on how structural damage changes driver-node demand in directed networks, with small interactive demos for pruning behavior.
Question
How many control inputs are needed to steer a directed network once edges start disappearing? This note focuses on the structural side of the question: how network topology alone shapes controllability and observability.
Key Idea
The framework uses maximum matching as a proxy for structural controllability. When fewer vertices can be matched after pruning, more driver nodes are required to control the full system.
Demonstration
These visuals are meant as intuition-builders rather than exact reproductions of the full workflow. The first block shows how pruning changes the current matching and the implied driver nodes; the second gathers small canonical motifs whose extraction curves summarize different controllability regimes.
- How a maximum matching highlights which targets still need direct control
- How pruning edges changes matching size and increases driver-node demand
- How simple motif families produce distinct cardinality profiles generation by generation
Matching Story
This row mirrors the matching story from the analysis code: start from the directed network, move to its bipartite representation, then return to the graph with unmatched targets highlighted. The pruning control removes edges from this toy network and updates the required driver nodes from the current maximum matching. You can drag the graph nodes in the first and third panels.
Cardinality curve
At each generation we remove one maximum matching and record its cardinality.
Cycle and chain share the same first-generation capacity in this extraction view.
We show chain separately, but it follows the same single-point curve as cycle.
Each round releases one match, so the profile decays as 1, 1, 1.
Convergent has the same extraction sequence as the divergent motif.
A denser graph starts with a larger extraction before tapering across later rounds.
Interpretation
The two blocks tell the same story at different scales. The matching view shows the immediate structural effect of edge loss on one concrete network: fewer usable matching edges means more unmatched targets and therefore more driver nodes. The cardinality motifs then compress that intuition into reusable patterns, showing which topologies exhaust quickly and which release control capacity over several generations.