Network Degree Calculator

Example data table

This example shows an undirected network using an edge list. The loop on node C illustrates loop handling options.

Formula used

• Undirected degree: \(k_i = \sum_j a_{ij}\), where \(a_{ij}\) is the adjacency value (1 if connected).
• Directed degrees: \(k_i^{in} = \sum_j a_{ji}\), \(k_i^{out} = \sum_j a_{ij}\), and \(k_i = k_i^{in}+k_i^{out}\).
• Weighted strength: \(s_i = \sum_j w_{ij}\), replacing 1 by edge weight \(w_{ij}\).
• Average degree: \(\langle k \rangle = \frac{1}{N}\sum_i k_i\). For simple undirected graphs, \(\langle k \rangle = \frac{2M}{N}\).
• Density: ratio of observed edges to possible edges (simple graphs only).

This calculator supports edge lists, adjacency lists, and adjacency matrices, and reports distributions \(P(k)\) as counts divided by \(N\).

How to use this calculator

1. Select graph type (directed or undirected).
2. Select edge weights. Use weighted mode if you provide weights.
3. Choose an input format, then paste your network data.
4. Set how to treat self-loops and whether to allow multi-edges.
5. Press Calculate. Results appear above the form.
6. Use Download CSV or Download PDF to export tables.

Tip: For directed networks, edges are interpreted as source target.

Network degree guide

1) Why degree matters in physical networks

Many physics systems can be modeled as graphs: atoms connected by bonds, contacts in granular media, power-grid buses, or nodes in transport and diffusion networks. Degree counts how many interactions a node has. High-degree nodes often act as strong couplers, accelerating energy exchange, synchronization, or spreading processes. In experiments, degree also serves as a quick proxy for local coordination.

2) Interpreting undirected degree

For undirected connections, the degree k is the number of neighbors of a node. If a network has N nodes and M edges, a key identity for simple graphs is the mean degree: <k> = 2M/N. For example, N = 100 and M = 250 gives <k> = 5. This average is a baseline for comparing different samples, lattice sizes, or experimental conditions.

3) Directed graphs: in, out, and total degree

Directed links represent asymmetric interactions such as flows, causal influence, or transitions in a Markov network. This calculator reports in-degree (incoming edges), out-degree (outgoing edges), and total degree. A node with large out-degree can be a strong source; large in-degree indicates a strong sink. Comparing the in/out balance helps reveal bottlenecks and preferred directions of transport.

4) Weighted strength adds interaction intensity

In weighted mode, each connection carries a weight (conductance, rate, capacity, or coupling). The calculator reports strength s, which sums weights incident to a node. Strength helps distinguish “many weak links” from “few strong links,” which can change transport, stability, and percolation thresholds. When weights vary widely, strength may correlate with load better than degree.

5) Degree distribution and hub detection

The distribution P(k) summarizes heterogeneity across nodes. Narrow distributions resemble random mixing, while heavy tails indicate hubs. Hubs can dominate dynamics: a small set of nodes may control diffusion time scales, synchronization onset, or failure cascades. The “Top hubs” table highlights the highest-degree nodes so you can target them for inspection, reinforcement, or further modeling.

6) Density, isolated nodes, and sparsity

Density is the fraction of possible edges that exist (shown for simple graphs). Sparse networks often have density near zero, even for large systems, yet can remain highly connected through a giant component. Isolated nodes (degree 0) can indicate disconnected sensors, missing measurements, or boundaries in a physical lattice. Tracking isolated counts across datasets is useful for quality control.

7) Self-loops and multi-edges in experiments

Self-loops may arise from coarse-graining, feedback, or bookkeeping conventions. You can ignore loops, count them as 1, or count them as 2 in undirected settings. Multi-edges represent repeated interactions, parallel channels, or time-aggregated contacts. Disabling multi-edges merges duplicates to approximate a simple graph, while enabling them preserves repeated events in temporal data.

8) Practical workflow for reliable results

Start by pasting an edge list from measurements or simulations. Verify node labels, then choose directed/undirected and weighted/unweighted modes. Compute results, inspect isolated nodes and hubs, and export tables for reporting. For comparisons across trials, track N, M, <k>, max degree, and P(k). If you change loop or multi-edge settings, report those choices to keep results reproducible.

FAQs

1) What is a node’s degree?

A node’s degree is the number of connections it has to other nodes. In undirected networks it counts neighbors; in directed networks it splits into in-degree and out-degree.

2) When should I use the weighted option?

Use weighted mode when each connection has an intensity, such as conductance, transition rate, coupling strength, or capacity. The calculator then reports node strength by summing those weights.

3) How are self-loops handled?

You can ignore loops, count them as one, or count them as two in undirected graphs. In directed graphs, a loop contributes one to both in-degree and out-degree.

4) Why does density show “N/A” sometimes?

Density is defined for simple graphs where each pair of nodes has at most one edge. If multi-edges are allowed, the calculator avoids reporting density because the “possible edges” count no longer matches.

5) What does the degree distribution table mean?

The table lists each degree value k, the number of nodes with that degree, and P(k) = count/N. It summarizes how evenly connections are spread across the network.

6) How do I interpret the “Top hubs” list?

Top hubs are the highest-degree nodes. They often indicate central junctions, highly connected particles, or major routers. In many dynamics, hubs influence spreading speed, load, and robustness.

7) Can I paste an adjacency matrix with labels?

Yes. Provide a header row and header column for node names, then fill the matrix with numbers. Nonzero entries are edges; in weighted mode, the values act as weights.