Mean distance μ(G) = 2W/(n(n-1)). Average shortest path. μ = 1 for complete graphs. μ = (n+1)/3 for paths. Small-world networks: μ ∝ log(n). The most cited metric in network science.
How to Use
Select graph:
- μ: Mean distance
- W: Wiener
- vs diam: Compare
Small World
μ ∝ log(n): small-world. μ ∝ n: large-world (paths, chains). μ ∝ n^{1/d}: d-dimensional lattices. The scaling of μ with n reveals the network's dimensional structure.
Bounds
1 ≤ μ ≤ (n+1)/3 for trees. 1 ≤ μ ≤ n/2 for general. μ = 1 iff K_n. μ close to 1: dense, well-connected. μ close to n/2: sparse, path-like.
Step-by-Step Instructions
- 1Select graph.
- 2Compute W (Wiener).
- 3μ = 2W/(n(n-1)).
- 4Compare with diameter.
- 5Check small-world.