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.

Networksintermediate

Closeness centrality

Q: An internal communications team must pick one person to trial a new announcement channel, so that news reaches the whole organisation in as few relays as possible. Which measure fits the brief best?

Closeness, because it measures average shortest-path distance to everyone else. The brief is about relays, and relays are path length. Closeness is defined as the inverse of average shortest-path distance, so the highest scorer reaches everyone in the fewest steps. Degree only counts the first hop. Density is a whole-network property and cannot rank individuals at all.

Q: A network splits into a main body of 180 staff and an isolated group of 4 contractors who talk only to each other. Standard closeness is computed on the whole network. What happens?

Distances between the two parts are infinite, so the standard formula is undefined for every actor. Every actor in the main body has at least four unreachable actors, and every contractor has 180, so every summed distance contains infinity. Nothing is dropped automatically. The usual fixes are to compute within the largest component and label the result, or to use harmonic closeness, where unreachable pairs contribute zero.

Closeness centrality measures how near an actor sits to everyone else. Take the geodesic distance from that actor to each other actor, meaning the number of steps on the shortest path, add those distances up, and invert. The standard normalisation divides $n-1$ by the total, giving a score between zero and one where high means a short average distance. Closeness reads as reach. It identifies who can get information out, or hear about a problem, in the fewest hops. It breaks on disconnected networks, where at least one distance is infinite, so a harmonic form is used instead.

Try it yourself

Closeness centrality and reach

Closeness divides 5 by the summed shortest-path distance to everyone else, so a high score means a short average distance to the rest of the team. Cut a tie that splits the team and the standard form stops being defined at all.

Closeness of Ana0.625

5 divided by a summed distance of 8 steps.

Circle size and shade both show closeness, 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 sits 8 steps from the other five in total, so closeness is 5 divided by that, 0.625. Ana and Dan hold the top score at 0.625, which is the shortest average distance to the rest. In steps that is the fewest relays, so Ana and Dan are the actors you would pick to spread news across the team. On steps the score is capped at one, which would mean being one step from everybody. Switch off Ana-Dan, Dan-Eve and Eve-Finn and the whole column stops being defined, because the team splits and at least one distance becomes infinite.

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

Closeness is a commute time. Someone near the centre of a city can get anywhere in twenty minutes. Someone at the end of a branch line can reach the same places, but every trip takes an hour. Both are on the network. Only one will hear about a problem on the day it happens. Closeness asks how far you sit from the average person, not how many people sit next to you.

Before you read on — recall

An internal communications team must pick one person to trial a new announcement channel, so that news reaches the whole organisation in as few relays as possible. Which measure fits the brief best?

Formulas

Closeness centrality, normalised

C_C(i) = \frac{n-1}{\sum_{j \ne i} d(i,j)}

Here

d(i,j)

is the number of steps on the shortest path between

i

and

j

. The numerator is the smallest the sum could possibly be, which happens when actor

i

is one step from everybody, so the score tops out at one. This form assumes a simple undirected network. On a directed relation, out-closeness uses distances from

i

and in-closeness uses distances to

i

, and the two rank people differently.

Worked on the six-person team

C_C(\text{Ana}) = \frac{5}{1+1+1+2+3} = \frac{5}{8} \approx 0.63

Ana is one step from Ben, Cara and Dan, two from Eve through Dan, and three from Finn. Finn sits at the far end with distances 3, 4, 4, 2 and 1, which sum to 14, so Finn scores 5 over 14, about 0.36. Dan, with only two contacts, also sums to 8 and ties Ana at the top.

Harmonic closeness for disconnected networks

C_H(i) = \frac{1}{n-1} \sum_{j \ne i} \frac{1}{d(i,j)}

An unreachable actor contributes a distance of infinity, and one divided by infinity is zero, so the sum stays finite. This version can be computed on any network, including one that has already split into separate components.

Worked examples

Scenario

A hospital wants to choose two staff to pilot a new handover procedure, so that word of the change spreads quickly and feedback comes back fast.

Solution

Degree would pick the two people with the most contacts, who are often in the same busy unit and reach the same colleagues twice over. Closeness ranks people by average distance to everyone, and in the running six-person team it ties Ana and Dan at about 0.63 even though Ana has three contacts and Dan has two. Dan scores high because Dan sits between the two halves, so nobody is far away. Taking the top two scores separately does not guarantee the best pair, because two high scorers can cover much the same colleagues. If joint reach is the goal, score candidate pairs on how many staff they reach together.

Scenario

An analyst computes closeness on a 400-person collaboration network, finds every score between 0.31 and 0.38, and reports a ranked leaderboard of the top ten.

Solution

The leaderboard is not trustworthy. In a large connected network almost everyone is a few steps from almost everyone, so closeness compresses into a narrow band and small measurement errors reorder the ranking. Two sensible responses. Report bands rather than ranks, or use a measure with more spread such as betweenness. It is also worth checking that the compression is not caused by a few hub accounts, such as an all-staff mailing list, that connect to everybody.

Common mistakes

✗Closeness is degree computed more slowly. Degree looks one step out and closeness looks at the whole network. An actor can have the most contacts and still sit at the edge, and an actor with two well-placed contacts can have the shortest average distance to everyone.
✗A higher closeness score means faster spread of anything. Closeness assumes whatever travels takes the shortest path and moves one step at a time. Money, gossip and infection do not all behave that way, and a measure built on shortest paths can mislead about a flow that wanders.
✗Closeness can be computed on any network as it stands. The standard formula breaks the moment one actor cannot reach another, because that distance is infinite. Either restrict the calculation to a single component and say so, or use the harmonic version.
✗A narrow range of closeness scores means everyone is equally placed. Compression is a property of the measure in large connected networks, not evidence about the organisation. It is a reason to distrust fine-grained rankings, not a finding.

Revision bullets

•Closeness inverts the summed shortest-path distance to all other actors
•Normalised form divides n-1 by that sum, so the maximum is one
•Reads as reach and speed of access, not popularity
•Undefined on disconnected networks; use harmonic closeness instead
•Running example: Ana and Dan both score 5 over 8, Finn scores 5 over 14
•Scores compress in large networks, so fine rankings are fragile

Quick check

An internal communications team must pick one person to trial a new announcement channel, so that news reaches the whole organisation in as few relays as possible. Which measure fits the brief best?

A network splits into a main body of 180 staff and an isolated group of 4 contractors who talk only to each other. Standard closeness is computed on the whole network. What happens?

Connected topics

Degree Centrality Betweenness Density Bridges and Cliques Social Analytics

More in Networks

What SNA Is Actors and Ties Weighted Ties Network Study

Sources

Sabidussi (1966)
Sabidussi, G. "The Centrality Index of a Graph." Psychometrika, 31(4), 581-603, 1966.
Defines centrality from the sum of distances to all other points, the basis of closeness.
Freeman (1978/79)
Freeman, L. C. "Centrality in Social Networks: Conceptual Clarification." Social Networks, 1(3), 215-239, 1978/79.
Gives the normalisation used here and separates closeness from degree and betweenness.
Boldi & Vigna (2014)
Boldi, P., & Vigna, S. "Axioms for Centrality." Internet Mathematics, 10(3-4), 222-262, 2014.
Argues for harmonic centrality as the well-behaved alternative when a network is not connected.

How to cite this page

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