Calculator Inputs
Formula Used
Betweenness centrality quantifies how often a node lies on shortest paths between other nodes. For node v:
This calculator uses the Brandes algorithm to compute betweenness efficiently for unweighted (BFS) and weighted (Dijkstra) graphs. Normalized values scale results by the maximum possible betweenness for the graph size.
How to Use This Calculator
- Paste your network as an edge list, one edge per line.
- Enable directed edges if links have a physical direction.
- Enable weights if edge costs differ; use positive weights.
- Click Calculate to display ranked centrality results.
- Download the table as CSV or PDF for reporting.
Example Data Table
A B B C C D B D D E
- B mediates paths from A to the rest.
- D bridges routes toward E.
- Nodes with low scores are less critical connectors.
Betweenness Centrality in Physical Networks
1) Why this metric matters
Many physical systems can be represented as graphs: atoms linked by bonds, pores connected by throats, power-grid components joined by lines, or transport corridors that carry particles, heat, or information. Betweenness centrality highlights nodes that frequently sit on shortest paths, making them likely mediators of flow redistribution when the system is stressed.
2) Shortest paths and effective transport
In diffusion and conduction problems, a “shortest path” can approximate an efficient route when edges represent travel time, resistance, or energetic cost. When weights are enabled, the calculator treats each edge weight as a cost, so lower total cost paths dominate the path counting used in the betweenness score.
3) Directed versus undirected structure
Directionality is essential for irreversible processes, active matter, pumping, or biased transport. A directed link u → v allows paths that follow the physical orientation, while excluding reverse motion. Comparing directed and undirected results helps separate geometric connectivity from dynamical constraints.
4) Weight choice and physical meaning
Choose weights that match your model: resistance for electrical networks, travel time for advection, inverse conductivity for thermal links, or free-energy barriers for hopping on landscapes. Because only positive weights are allowed, the computed shortest paths remain well-defined and numerically stable.
5) Interpreting high-betweenness nodes
A node with high betweenness is a potential bottleneck or bridge. In porous media, it may correspond to a key junction controlling percolation. In a lattice of coupled oscillators, it can indicate a mediator that couples distant regions. In networks of interactions, it marks a point where many optimal routes converge.
6) Normalization and cross-system comparison
Raw betweenness depends on network size. The normalized value rescales scores by the maximum possible betweenness for a graph with the same number of nodes, making it easier to compare different experiments, simulations, or snapshots of an evolving structure.
7) Endpoints option for specific analyses
The standard definition excludes endpoints because they are sources and targets, not intermediaries. Some workflows include endpoints to measure overall participation in shortest-path traffic. Use the endpoint option only if your interpretation explicitly treats origins and destinations as contributors.
8) Practical workflow and reporting
Start with a clean edge list, validate node labels, then compute centrality for unweighted and weighted cases. Inspect the top-ranked nodes as candidate control points for sensor placement, redundancy planning, or fault testing. Export results to CSV or PDF to document decisions and compare parameter sweeps consistently.
FAQs
1) What input format does the calculator accept?
Enter one edge per line as “u v”. For weighted analysis, use “u v w” with a positive numeric weight. Commas or spaces work, and lines starting with “#” are ignored.
2) What does a higher betweenness value mean?
It means the node lies on more shortest paths between other nodes. Such nodes often act as bridges or bottlenecks, so changes or failures there can strongly affect overall connectivity and transport efficiency.
3) When should I enable directed edges?
Enable direction when interactions have an orientation, such as one-way transport, biased hopping, or flow constrained by valves. If the process is symmetric, keep the graph undirected.
4) How should I choose edge weights?
Use a cost that adds along a path, such as resistance, time, attenuation, or barrier height. Lower total cost paths are treated as “shorter” and therefore contribute more to the computed betweenness.
5) Why are my normalized scores near zero?
In dense or highly redundant networks, many alternative shortest paths exist, spreading traffic across nodes. Normalization also divides by a size-based factor, so large graphs often yield smaller normalized values.
6) What is the endpoint option used for?
Standard betweenness counts only intermediate roles and excludes endpoints. The endpoint option adds a contribution for being a source or target, useful when you interpret participation in shortest-path traffic more broadly.
7) Can this handle disconnected graphs?
Yes. Pairs in different components have no connecting path, so they do not contribute. Nodes that bridge components within the connected part typically show higher scores than isolated nodes.