An independent study reference written by Dr Phuc V. Nguyen. It is not official subject material — for assessment requirements always follow your subject outline and vUWS.

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Betweenness centrality

Q: A firm merges two departments. Six months later one analyst has betweenness far above everyone else while degree is spread evenly. The most useful reading is that

the two former departments are still largely separate and most of the shortest routes between them now run through the analyst. Betweenness climbs when the shortest routes between two groups concentrate on one actor, which is what happens when the groups still have almost no direct ties. Even degree means nobody has an unusual number of contacts, so the spike is structural rather than a popularity effect. The score locates where the connection is thin without proving that the analyst is the sole crossing point, because a second and longer route may exist. Establishing that would take a separate check of whether removing the analyst splits the network. Either way the merge has not integrated the two groups.

Q: Between two actors there are four equally short paths, and Priya lies on two of them. How much does that pair contribute to the betweenness score of Priya?

One half. Credit is shared across the shortest paths. The pair contributes the number of paths through Priya divided by the total number of shortest paths, which is two divided by four, or one half. Awarding a full point for every path an actor touches would let one pair inflate the score, and awarding nothing would ignore genuine partial control.

Betweenness centrality counts how often an actor sits on the shortest path between two others. For every pair of other actors, work out the shortest paths between them and ask what share of those paths runs through the actor in question, then add that share across all pairs. High betweenness marks a broker, someone who carries or at least witnesses the flow between parts of the network that are otherwise far apart. It points to where a connection is thin, though calling someone a single point of failure is a separate test. That claim needs an articulation point, an actor whose removal splits the network into more components. Betweenness and degree often disagree, and the disagreement is usually the interesting part.

Try it yourself

Betweenness centrality and brokerage

Betweenness adds up the share of shortest paths between other pairs that run through an actor. Each unordered pair is counted once and the score is left unnormalised, which is the form the concept page prints.

Betweenness of Ana6 pairs

Out of the 10 unordered pairs that exclude Ana. Normalised that is 0.60.

Circle size and shade both show betweenness, and the number under each name is that same figure, so nothing is carried by colour alone. Ana and Dan hold the highest score.

Size and shade the actors by

Path distance

Selected actor

Ties (6 of 15 on)

Ana-Ben12Ana-Cara10Ana-Dan2Ana-Eve5*Ana-Finn5*Ben-Cara9Ben-Dan5*Ben-Eve5*Ben-Finn5*Cara-Dan5*Cara-Eve5*Cara-Finn5*Dan-Eve3Dan-Finn5*Eve-Finn8

Weights are contact counts over a seven-day period. The six ties of the running team carry the recorded counts. The other nine pairs carry a placeholder of 5, marked with an asterisk, so they can be switched on at all. No placeholder tie is on, so every weight on screen is a recorded one.

Every actor, every measure, recomputed from the ties on screen. Σd is the summed shortest-path distance to the other five actors, in steps. Closeness is 5 divided by that sum. Harmonic is the normalised harmonic closeness, a different measure that averages 1 divided by each distance and treats an unreachable actor as a contribution of zero, with n held at 6 for the whole team rather than shrunk to a component. Betweenness counts each unordered pair once and is not normalised. Clustering is 2e divided by k(k−1) and needs at least two neighbours.
Actor	Degree	k / 5	Strength	Σd (steps)	Closeness	Harmonic	Betweenness	Clustering
Ana (selected)	3	0.60	24	8	0.625	0.767	6	0.33
Ben	2	0.40	21	11	0.455	0.617	0	1.00
Cara	2	0.40	19	11	0.455	0.617	0	1.00
Dan	2	0.40	5	8	0.625	0.700	6	0.00
Eve	2	0.40	11	10	0.500	0.633	4	0.00
Finn	1	0.20	8	14	0.357	0.467	0	—

The team is one component, so every distance is finite and standard closeness has a value for everyone. Clustering is an em dash for Finn, because a coefficient needs at least two neighbours to have any pair to check.

Ties 6 of 15 Components 1 Density 0.40 Standard closeness defined

Ana lies on the shortest path for 6 of the 10 pairs that exclude Ana, which is 0.60 once normalised. Ana and Dan hold the most brokerage at 6, while Ana holds the most ties at 3. Different names top the two measures, which is the disagreement worth showing a class. A high score locates a thin connection. It does not prove a single point of failure, and that separate claim needs an articulation point. Removing Ana, Dan and Eve would split the team, so those actors are where the structural risk actually sits.

Every figure here is recomputed from the ties currently switched on. Distances come from breadth-first search on steps, or from Dijkstra on 1 divided by the weight when weighted distance is selected, and betweenness comes from the Brandes accumulation over those same shortest paths. Betweenness counts each unordered pair once and is left unnormalised. Bridges and articulation points are found by removing the tie or the actor and recounting the components, not by pattern matching.

Why it matters

Think of a road map where every trip takes the shortest route. Some towns are only destinations. Others are the one place a bridge crosses the river, so half the traffic in the region passes through even though hardly anyone lives there. Betweenness counts that traffic. It is the difference between being popular and being unavoidable.

Before you read on — recall

A firm merges two departments. Six months later one analyst has betweenness far above everyone else while degree is spread evenly. The most useful reading is that

Formulas

Betweenness centrality

C_B(i) = \sum_{s < t,\; s,t \ne i} \frac{g_{st}(i)}{g_{st}}

Here

g_{st}

is the number of shortest paths between

s

and

t

, and

g_{st}(i)

is how many of them pass through

i

. If there is exactly one shortest path and it runs through

i

, that pair contributes one. If there are four shortest paths and

i

lies on one, the pair contributes a quarter.

Normalised betweenness

C_B'(i) = \frac{C_B(i)}{(n-1)(n-2)/2}

The denominator is the number of unordered pairs that exclude actor

i

, which is the most any actor could possibly broker. With six actors that is ten pairs, so a raw score of 6 normalises to 0.6. That form is for a simple undirected network. On a directed relation, ordered pairs count separately and the denominator becomes

(n-1)(n-2)

Worked on the six-person team

C_B(\text{Dan}) = 6, \qquad C_B(\text{Ana}) = 6, \qquad C_B(\text{Ben}) = 0

Dan lies on the only shortest path for six pairs: Ana-Eve, Ana-Finn, Ben-Eve, Ben-Finn, Cara-Eve and Cara-Finn. Ana brokers the mirror image, the six pairs joining Ben or Cara to Dan, Eve or Finn. Ben and Cara broker nothing, because every path through them has a shorter alternative through Ana. Dan matches Ana on brokerage while holding one fewer tie, so the two tie at the top and neither is uniquely the most critical actor. Both also happen to be articulation points here, which is the separate property that makes a removal split the team.

Worked examples

Scenario

A manufacturer maps which engineers consult which colleagues on production faults, and wants to know where a resignation would hurt most.

Solution

Ranked on degree, the head of the largest sub-team leads. Ranked on betweenness, a mid-level engineer leads, because that person is the only link between the plant floor group and the design office. When the engineer resigns, degree barely moves for anyone, but the two groups lose their shortest route and fault reports start taking seven days to reach design. The practical action is to build a second connection before the person leaves, not to promote them.

Scenario

A bank applies betweenness to its payments network to flag accounts that might be moving funds for someone else, on the theory that money passes through brokers.

Solution

That theory holds only if money takes shortest paths, and often it does not. Funds are deliberately routed through long chains precisely so they do not look direct. Betweenness assumes the thing flowing seeks the shortest route and travels as one indivisible parcel, which suits a rumour or a physical delivery better than a deliberately obscured chain. The honest step is to state the flow assumption, then either pick a walk-based measure or treat betweenness as one weak signal among several.

Common mistakes

✗High betweenness means high status. Brokers are frequently mid-level. The position comes from spanning a gap, and senior people usually sit inside a dense group of peers where every path has an alternative. Status is read better from in-degree on a directed relation.
✗Betweenness and degree measure roughly the same thing. They can disagree completely. In the six-person example Dan holds two ties and the same brokerage as Ana with three, because both of the ties Dan holds span a gap while one of the three Ana holds runs inside the Ana-Ben-Cara triangle. That team carries three bridges, Ana-Dan, Dan-Eve and Eve-Finn, so Dan sits on a thin link rather than on the only one. Networks with clear sub-groups routinely produce low-degree, high-betweenness actors.
✗A broker is always good for the organisation. A broker is a route and a bottleneck at the same time. The position that lets someone connect two groups also lets them filter, delay or distort what crosses. Whether that is valuable depends on whether you want control or resilience.
✗Betweenness is cheap to compute on any size of network. It needs shortest paths between all pairs, so the work grows with the number of actors multiplied by the number of ties. On very large networks it is usually estimated from a sample of source actors rather than computed exactly.

Revision bullets

•Counts the share of shortest paths between other pairs that run through the actor
•Normalise by (n-1)(n-2)/2, the number of pairs excluding that actor
•Identifies brokers and structural gaps; a true split needs an articulation point
•Can be high for a low-degree actor: Dan scores 6 on two ties in the running example
•Assumes flow takes the shortest path, which not every flow does
•Expensive to compute, so large networks use sampled estimates

Quick check

A firm merges two departments. Six months later one analyst has betweenness far above everyone else while degree is spread evenly. The most useful reading is that

Between two actors there are four equally short paths, and Priya lies on two of them. How much does that pair contribute to the betweenness score of Priya?

Connected topics

Degree Centrality Closeness Bridges and Cliques Network Study Analytics Leverage

More in Networks

What SNA Is Actors and Ties Weighted Ties Density

Sources

Freeman (1977)
Freeman, L. C. "A Set of Measures of Centrality Based on Betweenness." Sociometry, 40(1), 35-41, 1977.
The original definition, including the shared-credit rule when several shortest paths exist.
Burt (2004)
Burt, R. S. "Structural Holes and Good Ideas." American Journal of Sociology, 110(2), 349-399, 2004.
Evidence that people whose contacts span a structural gap generate ideas judged more valuable.
Borgatti (2005)
Borgatti, S. P. "Centrality and Network Flow." Social Networks, 27(1), 55-71, 2005.
Shows that each centrality assumes a particular kind of flow, and misfires when the assumption is wrong.
Brandes (2001)
Brandes, U. "A Faster Algorithm for Betweenness Centrality." Journal of Mathematical Sociology, 25(2), 163-177, 2001.
The algorithm behind most practical implementations, and the source of the cost estimate quoted here.

How to cite this page

Dr. Phil's Quant Lab. (2026). Betweenness centrality. Business Analytics Atlas. https://phucnguyenvan.com/analytics_atlas/concept/ba-betweenness-centrality