Hands-on 2 · Global Properties
Whole-graph measures summarise how the system is organised. But the number you get means little if not compared to a reference network.
Clustering C and local efficiency (segregation): how often a node's neighbours are connected to each other, and how well they still communicate if the node is removed. Path length L and global efficiency E (integration): how few steps separate nodes on average. Efficiency uses 1/distance, so it still works when the graph is fragmented.
Question. Is C = 0.5 high or low? Compared with what?
A reference graph has the same number of nodes and edges as ours but lacks the structure we want to test. Normalised measures divide each value by the average over many reference graphs. The reference you choose decides what “more clustered than expected” means.
A small-world graph is much more clustered than random (C/Crand ≫ 1) while its path length stays close to random (L/Lrand ≈ 1). The index σ = (C/Crand)/(L/Lrand) is usually called small-world when σ > 1. No single density is “right”, so a good practice is to compute the whole curve.
σ is defined against random graphs, so the lattice is not offered here. Compare the degree-preserving and correlation-based curves: the second keeps the transitivity of correlations.
HbO, HbR and HbT give different graphs.
Question. Which chromophore gives the least clustered graph? Choose the chromophore(s) to interpret before you look at results, and remember to report at least HbO and HbR.
Start from a ring where each node links to its nearest neighbours, then rewire each edge with probability p. A few random shortcuts collapse the path length while clustering stays high.
Merging channels into anatomical regions is a literal coarse-graining. Each region-to-region edge is the average Fisher-z of the channel pairs between the two regions. Random-graph references scale as Crand ≈ ⟨k⟩/N and Lrand ≈ ln N / ln⟨k⟩, so raw values change with N.