High-degree "rich" nodes preferentially connect to each other, forming an elite club. The rich-club coefficient ϕ(k) measures how densely hubs interconnect.
Network topology (size ∝ degree)
Rich-club coefficient ϕ(k) vs degree threshold k
The rich-club coefficient ϕ(k) = E(>k) / [N(>k)(N(>k)−1)/2]: the fraction of possible edges present among nodes with degree > k. Values above the null-model expectation indicate rich-club organization. Found in the internet backbone, brain connectome, airline networks, and scientific collaboration.