K-Core Decomposition Calculator

Inputs

Graph mode Undirected Directed (treated as undirected for coreness)

Most k-core definitions assume undirected graphs.

Delimiter Auto (space/comma/tab) Whitespace only Comma only

Use one edge per line: u v

Selected k (extract k-core)

Shows the subgraph induced by nodes with core ≥ k.

Options

Drop self-loops

De-duplicate edges

Recommended for messy datasets.

Edge list A B A C A D B C B D C D D E E F E G F G G H

Nodes may be names or numbers. Lines starting with # are ignored.

Calculate Load Example Clear

The k-core of a graph is the largest subgraph in which every node has degree at least k within that subgraph. K-core decomposition assigns each node a core number (coreness), defined as:

• Core number of node v is the maximum k such that v belongs to the k-core.
• The graph’s degeneracy is the maximum core number over all nodes.

This calculator uses an iterative peeling procedure: repeatedly remove the current minimum-degree node, update neighbor degrees, and record the degree at removal as the node’s core number.

1. Paste your edge list, one connection per line.
2. Choose the delimiter option that matches your data.
3. Set the selected k to extract a specific k-core subgraph.
4. Click Calculate to compute core numbers and shell counts.
5. Use Download CSV or Download PDF for reports.

1) Purpose of k-core decomposition

K-core decomposition summarizes how strongly a network can “hold together” under pruning. It repeatedly removes weakly connected nodes and reveals a resilient backbone. The main output is a core number for each node, plus the network’s degeneracy, which is the maximum core number observed.

2) What this calculator produces

Submit an edge list and the tool returns three practical views: a coreness table for all nodes, a shell distribution showing how many nodes fall into each k-shell, and the induced subgraph for a chosen k (nodes with core ≥ k) with its remaining edge list.

3) Data inputs and cleaning options

Each line describes one connection, such as u v or u,v. Auto delimiter handles spaces, tabs, and commas. Dropping self-loops prevents artificial degree inflation. De-duplicating edges helps when logs repeat interactions, keeping degree-based coreness closer to the underlying structure.

4) Algorithm used and computational cost

The calculation uses iterative peeling: remove the current minimum-degree node, update neighbor degrees, and record the degree at removal as the node’s core number. A priority queue accelerates selection, yielding typical behavior near O((n+m) log n) for n nodes and m edges in sparse graphs.

5) Interpreting core numbers as measurements

If a node has core number 4, it belongs to the 4-core but not the 5-core. Higher values indicate stronger mutual support under pruning, not just high raw degree. The shell distribution adds context: a large low-k mass suggests a periphery, while substantial high-k shells signal a robust nucleus.

6) Choosing k for comparison studies

Start with small k (1–2) to remove isolated noise while keeping most nodes. Increase k to focus on stable cores. Compare the remaining node and edge counts at the same k across conditions (before/after, control/treatment, different timesteps) to quantify changes in cohesion.

7) Physics and complex-systems relevance

In network physics, k-cores help describe robustness and threshold-like activation. Dense cores can approximate backbones in mutual reinforcement dynamics and resilience under failures. For contact or interaction networks, high-k nodes often correspond to persistent, strongly supported structures that survive pruning of weak links.

8) Reporting with exports

For reproducible reporting, record total nodes, total edges, degeneracy, and your chosen k, plus the extracted k-core size (nodes/edges). Export CSV to archive coreness labels and shell counts for analysis pipelines. Export a PDF snapshot for sharing, teaching, or quick documentation.