Reveal hubs that preferentially connect with peers directly. Evaluate rich-club curves across thresholds quickly here. Use exports and examples to validate your network insights.
The rich-club coefficient measures how densely “rich” nodes connect among themselves. Richness can be degree k or strength s (sum of weights).
Unweighted rich-club coefficient for an undirected network at threshold k:
φ(k) = 2E>k / (N>k(N>k − 1))
Weighted rich-club coefficient uses the total weight inside the rich subgraph:
φw(t) = Wrich / ∑i=1..Erich w(i)
Normalized rich-club coefficient compares observed φ to randomized networks: ρ(k) = φ(k) / φrand(k) where random networks preserve the degree sequence via edge rewiring.
This example illustrates a small undirected weighted network. Load it using “Load Example”.
| Edge list line | Meaning |
|---|---|
| A B 3 | Strong connection between A and B |
| A C 2 | Moderate connection between A and C |
| B C 2 | Moderate connection between B and C |
| B D 1 | Weaker tie between B and D |
| C D 1 | Weaker tie between C and D |
| D E 1 | Peripheral node E connected to D |
For threshold 1, rich nodes are A, B, C, D; φ increases when these nodes are densely linked.
Many physical and engineered systems form networks: grids, transport, climate links, and interaction graphs. A rich club appears when high‑richness nodes connect to each other more than expected, creating a backbone that can speed diffusion, support synchronization, or concentrate vulnerability.
For each threshold, the calculator selects nodes with richness greater than the threshold and counts links among them. It reports Nrich, Erich, and the density φ. In undirected graphs, φ ranges from 0 to 1, where 1 indicates a fully connected rich subgraph.
Degree tracks neighbor count and highlights topological hubs. Strength sums weights and reflects interaction intensity such as coupling, capacity, or correlation. Comparing both views can show whether “elite” organization is structural, weight‑driven, or a combination of the two.
Rich‑club analysis is best viewed as a curve across thresholds rather than a single point. As the threshold rises, Nrich usually shrinks, so interpret φ together with club size. A persistent rise of φ at higher thresholds often indicates a stable hub‑to‑hub core.
The weighted coefficient φw compares Wrich to the sum of the strongest Erich weights available in the entire network. Values near 1 suggest rich nodes capture a large share of the strongest interactions, not just many links.
Dense hub subgraphs can arise simply because high‑degree nodes have many pairing options. Normalization estimates φrand using randomized networks that preserve each node’s degree via edge rewiring. The ratio ρ = φ/φrand highlights excess rich‑club structure; ρ > 1 over a threshold band is a common signal.
The tool ignores self‑loops and can merge duplicate undirected edges by summing weights or taking the maximum. These choices influence strength and Wrich. Ensure delimiters match your input, and keep thresholds where at least two rich nodes exist, otherwise estimates become unstable.
When comparing systems or conditions, use identical preprocessing, threshold steps, and weight scaling. Report Nrich and Erich alongside φ, and include ρ when normalized. Exported tables make it easy to plot curves and document parameter choices for reproducibility.
It means rich nodes are very densely connected among themselves. In an undirected network, φ = 1 implies the rich subgraph is complete, with every rich node connected to every other rich node.
High‑degree nodes naturally have many connection opportunities, which can inflate density. That is why comparing against degree‑preserving random networks and examining ρ is valuable for interpretation.
Use strength when edge weights represent meaningful physical intensity, capacity, or coupling. Strength can reveal a core formed by strong interactions even if the degree‑based rich‑club signal is weak.
A practical range is from 0 up to the maximum observed richness, using step 1 for degree or a larger step for strength. Avoid thresholds that leave fewer than two rich nodes, because φ becomes uninformative.
The calculator sums weights inside the rich subgraph (Wrich) and divides by the sum of the globally strongest Erich edge weights. This highlights whether the rich club captures top‑weight interactions.
The implemented randomization uses degree‑preserving edge swaps for undirected graphs. Extending this to directed or weighted constraints requires different null models and can change what “preserved structure” means.
Always report Nrich and Erich for each threshold, and include φrand and ρ when normalized. These values explain whether changes in φ reflect real densification or simply shrinking club size.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.