Hubs, modules and resilience

The degree of a node is its number of edges. Random graphs have a narrow, Poisson-like degree distribution; many real networks have heavy tails (hubs). With about 100 nodes, power-law claims cannot be tested.

Strength: sum of edge weights. Betweenness: fraction of shortest paths through the node (a bridge). Eigenvector: connected to other well-connected nodes. The five most central channels for each measure are circled.

Question. Are the same channels central for all measures? Which would you call a hub? In fNIRS, check whether a “hub” is simply a channel with strong residual systemic signal.

Modularity Q compares the weight inside modules with the weight expected by chance for the same node strengths. The Louvain algorithm maximises Q greedily and finds the number of modules; it is stochastic, so each run can give a different partition, and the resolution γ controls module size.

Spectral clustering optimises something else (a normalised cut) and needs the number of modules k chosen in advance: ask for k modules and you get k, whether or not they are there. Compare the two on the same graph.

Question. Do modules follow lobes? Hemispheres? Or simply neighbouring channels that share optodes and scalp?

The participation coefficient P measures how evenly a node's edges spread across modules (0: all inside its own module). The within-module degree z measures how central it is inside its module. High z and low P: a provincial hub. High z and high P: a connector hub.

Remove channels one by one, either at random (failure) or always the one with the highest current degree (targeted attack on hubs), and recompute after each step. The dashed line repeats the attack on degree-preserving random graphs. Uses the signal and density from step 1.

Question. Which hurts the network more? Why does the fNIRS graph resist the attack better than a random graph with the same degrees? What would losing a cluster of optodes (poor contact) do?