Extra connectivity κₖ(G): minimum vertices to remove leaving all components with >k vertices. Hierarchy: κ₀ = κ ≤ κ₁ ≤ κ₂ ≤ .... Higher k = stronger resilience requirement. For hypercubes: κₖ = (k+1)(n-k)-k for small k.
How to Use
Select graph and k:
- κₖ: k-extra conn.
- k: Min comp. size
- Hierarchy: κ₀≤κ₁≤...
Connectivity Hierarchy
κ₀ = standard connectivity. κ₁ = super-connectivity (no isolates, components ≥ 2). κ₂ = all components ≥ 3. Each level reveals finer resilience structure. Networks should aim for high κₖ for critical k.
Hypercubes
Q_n (n-dimensional hypercube): κₖ = (k+1)(n-k)-k for 0 ≤ k ≤ n-3. Beautiful closed form! Shows hypercubes have excellent higher-order resilience. Explains their use in parallel computing.
Step-by-Step Instructions
- 1Select graph.
- 2Choose k.
- 3Compute κₖ.
- 4Build hierarchy.
- 5Compare with design.