Reliability R(G,p): probability that G stays connected when each edge independently works with probability p. R(K_n,p) and R(tree,p) have closed forms. R is a polynomial in p of degree |E|. Higher connectivity → higher reliability.
How to Use
Select graph:
- R(p): Reliability
- Curve: R vs p
- Compare: Topologies
Network Design
Goal: maximize R(G,p) given constraints (edges, vertices). More connections = higher reliability but higher cost. Optimal: expander graphs, Ramanujan graphs. Uniformly most reliable graph for fixed n,|E|: open problem!
Computation
#P-hard to compute (Valiant, 1979). But R = Tutte evaluation, so deletion-contraction works. Inclusion-exclusion over cuts. Monte Carlo estimation for large networks. Important in telecommunications.
Step-by-Step Instructions
- 1Select graph.
- 2Set p.
- 3Compute R(G,p).
- 4Plot curve.
- 5Compare topologies.