Calculate global or local clustering coefficient for a simple, undirected, unweighted graph from triangles, connected triplets, node degree, neighbor links, or a degree sequence.
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Clustering Coefficient Formula
Local clustering coefficient for a node with degree k and e links among its neighbors:
Global clustering coefficient (transitivity) for the whole network:
Connected triplets from a degree sequence:
- k — degree of the node (number of neighbors)
- e — actual edges between that node’s neighbors
- k(k − 1)/2 — maximum possible edges among k neighbors
- Triangle — three nodes all connected to each other
- Connected triplet — any node with two distinct neighbors (open or closed)
Each triangle contains three closed triplets, which is why the global formula multiplies triangles by three. The local formula requires k ≥ 2, since a node with one or zero neighbors has no possible neighbor pairs. Both coefficients return values between 0 and 1 when the denominator is positive. Degree sequences are checked for graphical validity, but aggregate checks do not prove that a supplied triangle count is realizable. Global transitivity is not the average of local coefficients.
Reference Values
Network type alone does not determine a universal clustering range:
| Network Type | Clustering context |
|---|---|
| Random graph (Erdős–Rényi) | ≈ p (edge probability) |
| Social networks | Depends on graph and definition |
| Collaboration networks | Depends on graph and definition |
| Web / hyperlink graphs | Directed hyperlink graphs require a specified directed definition |
| Biological (protein interaction) | Depends on graph and definition |
| Lattice / regular graph | Depends on the lattice and neighborhood |
The numeric ranges below describe fractions, not universal quality categories:
| Value | Interpretation |
|---|---|
| C = 0 | No supplied triplets are closed; locally, no neighbor links exist |
| 0 < C < 0.3 | More than 0% and less than 30% |
| 0.3 ≤ C < 0.6 | At least 30% and less than 60% |
| C ≥ 0.6 | At least 60% of the relevant denominator |
| C = 1 | Locally, every neighbor pair is linked; globally, every connected triplet is closed |
Worked Examples
Example 1 — Local coefficient. A node has 4 neighbors and 3 links among those neighbors.
- Possible neighbor links = 4 × 3 / 2 = 6
- C_i = 3 / 6 = 0.5000
Example 2 — Global coefficient. A graph has 12 triangles and 90 connected triplets.
- Closed triplets = 3 × 12 = 36
- C = 36 / 90 = 0.4000
Example 3 — From a degree sequence. Degrees [2, 3, 3, 1, 1] with 1 triangle.
- Triplets = 1 + 3 + 3 + 0 + 0 = 7
- C = 3 / 7 ≈ 0.4286
FAQ
What is the difference between local and global clustering? Local C measures one node’s neighborhood. Global C (transitivity) summarizes the whole network in a single number.
Why does my node need degree ≥ 2? A node with one neighbor has no neighbor pairs, so the denominator k(k − 1)/2 is zero and the coefficient is undefined.
Triangles or closed triplets — which should I enter? Either works. The calculator multiplies triangles by 3 internally to get closed triplets. Use whichever count you already have. A whole-graph closed-triplet count must be a multiple of three.
Can the coefficient exceed 1? No. If you get a value above 1, your closed-triplet or triangle count is inconsistent with your triplet count.
