Clustering Coefficient Calculator

Formula used

In network physics and complex systems, the clustering coefficient quantifies how often a node’s neighbors also connect. It highlights local triangle structure, which affects diffusion, percolation, synchronization, and resilience.

Undirected local clustering

For node i with degree k_i, let E_i be the number of edges among its neighbors. The local clustering coefficient is:

Cᵢ = 2Eᵢ / (kᵢ (kᵢ − 1))

The average clustering is the mean of Cᵢ over nodes with kᵢ ≥ 2.

Global transitivity (undirected)

Transitivity measures triangle density relative to connected triplets:

T = 3 × (number of triangles) / (number of connected triplets)

Directed local clustering (advanced)

In directed mode, the calculator uses a directed, triangle-based local clustering formulation. It accounts for different triangle orientations using a symmetrized adjacency approach. A symmetrized transitivity is also reported for an intuitive global view.

How to use this calculator

1. Choose Undirected for standard clustering in interaction networks.
2. Paste your edge list, one connection per line (e.g., A B).
3. Use consistent labels for nodes; spaces or commas both work.
4. Click Calculate to see summary values and per-node results.
5. Use Download CSV or Download PDF to export the table.

Example data table

Example undirected edge list forms two triangles sharing node C. Expected values are shown for quick verification.

A B
B C
C A
C D
D E
E C

Article

1) Why clustering matters in physical networks

Clustering coefficient measures how often neighbors of a node are also connected, forming triangles. In physics, triangles indicate local cohesion in interaction graphs, transport pathways, and correlation structure in complex systems. High clustering can slow spreading fronts yet stabilize local synchronization, depending on coupling strength and topology.

2) Local vs global viewpoints

This calculator reports per-node local clustering and summarizes behavior using average clustering and transitivity. Local values highlight heterogeneous regions, while transitivity captures overall triangle density relative to connected triplets. Comparing both helps distinguish a few highly clustered hubs from broadly clustered networks.

3) Interpreting degree effects

For undirected graphs, nodes with degree 0 or 1 have undefined triangle opportunities, so their local clustering is treated as zero for display. As degree increases, the number of possible neighbor connections grows as k(k−1)/2, so even modest triangle counts can yield small clustering in high-degree regions.

4) Triangles, motifs, and physical interpretation

Triangles are a basic motif linked to redundancy and local feedback. In lattice-like contact networks, triangles indicate short loops that enhance local return probability for random walks. In polymer or soft-matter contact graphs, triangles can reflect compact packing and constrained rearrangements.

5) Directed interactions and asymmetry

Many systems are directional: information flow, causal influence, or driven transport. Directed clustering evaluates closed triplets while accounting for arrow orientation. The directed option here uses a triangle-based formulation built from a symmetrized adjacency view, then reports an intuitive symmetrized transitivity for global comparison.

6) Data entry and reproducibility

Provide an edge list with one connection per line using labels such as A, node_12, or 3. Keep labeling consistent across runs. For reproducible studies, export CSV for analysis notebooks and PDF for reports, ensuring the same direction setting and self-loop choice are recorded.

7) Practical ranges and sanity checks

Clustering values lie between 0 and 1. A complete clique yields 1 for every node. A tree has 0 triangles, giving zero clustering and transitivity. Small-world networks often show higher clustering than random graphs with similar degree, while scale-free networks can show mixed local clustering by degree.

8) Using results in research workflows

Use node-level clustering to identify mesoscale structure, then compare average clustering and transitivity across conditions, time windows, or parameter sweeps. When studying dynamics, relate clustering to diffusion times, epidemic thresholds, or synchronization onset. Combine clustering with path length and assortativity for a fuller structural picture.

FAQs

1) What does a clustering coefficient of 0 mean?

It means there are no triangles around that node or across the network. Neighbors are not interconnected, so local neighborhoods look like open chains rather than closed loops.

2) Why are degree 0 or 1 nodes shown with zero clustering?

With fewer than two neighbors, a node cannot form a triangle, so the denominator k(k−1) becomes zero. The calculator displays 0 to keep tables consistent and exports usable.

3) What is the difference between average clustering and transitivity?

Average clustering is the mean of local coefficients (typically over nodes with k≥2). Transitivity is a global ratio based on triangles versus connected triplets, weighting higher-degree nodes more strongly.

4) How should I format my edge list?

Enter one edge per line as “u v” or “u,v”. Labels can be text or numbers. Optional third values are ignored. Use the same label spelling everywhere to avoid accidental duplicate nodes.

5) When should I use directed mode?

Use directed mode when links represent asymmetric influence or flow, such as causal interactions, transport direction, or regulatory edges. It evaluates triangle closure while considering arrow orientation and also reports a symmetrized transitivity for context.

6) Do self-loops affect clustering results?

Self-loops rarely represent physical neighbor connections and can bias local counts. If enabled, they are treated as edges from a node to itself, which may alter degree-like quantities and triangle terms in directed mode.

7) How can I verify the calculator is working correctly?

Load the built-in example and compare your output to the example table. For additional checks, test a triangle (three nodes fully connected) which should give local clustering 1 for each node.