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

Degree centrality

Q: Two employees each have a degree of 6 in the same advice network. One sits inside a single tight project team, the other has one contact in each of six departments. What follows?

Degree ranks them equally, but their reach and brokerage positions can differ sharply. Degree counts neighbours and stops there. Both have six, so degree cannot separate them. The employee spanning six departments will usually score higher on closeness and betweenness, because those measures look at paths beyond the first step, though how much higher depends on how well connected those six departments already are. Equal degree with unequal position is the standard reason to report more than one centrality.

Q: A consultancy reports that Team A is better connected than Team B because average degree is 7.2 against 4.1. Team A has 12 members and Team B has 90. The comparison is

misleading, because the ceiling on degree is 11 in Team A and 89 in Team B. A 12-person team cannot exceed a degree of 11, so 7.2 is about 65 per cent of the maximum. In a 90-person team the ceiling is 89, so 4.1 is under 5 per cent. Raw average degree therefore sets two numbers with different ceilings side by side, which is what makes the claim misleading. Normalising strips out the ceiling and does no more than that. Calling Team A better connected would still need both studies to draw the boundary the same way and measure the same relation, and it would need a benchmark for groups of each size. Matching the relation on its own does not repair the comparison, because the ceiling still differs.

Degree centrality counts the ties an actor has. It is the simplest statement of position and often the most useful, because it captures direct activity and immediate exposure. Dividing by the largest number of ties anyone could have, which is $n-1$ , gives a normalised score between zero and one. That removes the mechanical size ceiling and nothing else, so comparing two networks still needs matched boundaries, the same relation and a size benchmark. On a directed relation, degree splits in two. In-degree counts ties received and usually reads as prestige or demand. Out-degree counts ties sent and usually reads as activity or effort. Degree is entirely local, so it sees the neighbourhood and nothing beyond it.

Try it yourself

Degree centrality

Degree counts the ties an actor holds, and normalised degree divides that by the 5 other actors. It is entirely local, so it sees the neighbourhood and nothing past it. Toggle a tie and only the two actors it touches change degree, while the path-based columns can move for everyone.

Degree of Ana3 ties

Normalised, 3 divided by 5 = 0.60. The ceiling of 5 is the number of other actors.

Circle size and shade both show degree, and the number under each name is that same figure, so nothing is carried by colour alone. Ana holds 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 holds 3 ties, which is 0.60 once divided by the 5 other actors. Degree stops at the first step, so it cannot tell you where those ties lead. The two measures disagree here. Ana leads on degree with 3, while Ana and Dan broker the most pairs at 6. That gap is the reason a network study reports more than one centrality. Strength happens to put the same name on top here, Ana on 24 contacts, because at this setting the heaviest ties and the most ties sit with the same person.

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

Degree is the headcount of your contacts. It answers one question well. If something starts with you, how many people does it reach in a single step. It says nothing about where those contacts then lead. A person with eight contacts who all know each other reaches eight people and stops. A person with three contacts in three different departments may reach the whole organisation.

Before you read on — recall

Two employees each have a degree of 6 in the same advice network. One sits inside a single tight project team, the other has one contact in each of six departments. What follows?

Formulas

Degree centrality

C_D(i) = k_i = \sum_{j \ne i} a_{ij}

Add up the adjacency row for actor

i

. In the running six-person team Ana is tied to Ben, Cara and Dan, so her degree is 3, while Finn is tied only to Eve, so his degree is 1.

Normalised degree centrality

C_D'(i) = \frac{k_i}{n-1}

The denominator is the number of other actors, which is the most ties anyone could hold. With six actors the denominator is five, so Ana scores 0.6 and Finn scores 0.2. Normalising strips out the size ceiling and does no more than that. Mean normalised degree in a simple undirected network is exactly its density, which the density node shows still falls with size, so matched boundaries, a matched relation and a size benchmark are what license a comparison. On a directed relation, normalise in-degree and out-degree separately, each by

n-1

In-degree and out-degree

k_i^{\text{in}} = \sum_{j} a_{ji}, \qquad k_i^{\text{out}} = \sum_{j} a_{ij}

Sum the column for ties received and the row for ties sent. In an advice network a high in-degree marks the person others rely on and a high out-degree marks the person who needs help. Reporting only the total hides which of the two you found.

Worked examples

Scenario

An operations manager wants to find the person whose absence would slow the team most, and starts by ranking everyone on degree in the running advice network.

Solution

Degree puts Ana first with three contacts. Ben, Cara, Dan and Eve tie on two, and Finn is last with one. The ranking is defensible and incomplete. Dan also has two contacts, yet Dan is the only route between the Ana-Ben-Cara group and the Eve-Finn side. If Dan takes leave the team splits in half. If Ben takes leave nothing disconnects. Degree cannot see that, because it never looks past the immediate neighbours. That is the gap betweenness fills.

Scenario

A support desk logs which agent escalated which ticket to which colleague over a quarter. Management wants to identify subject-matter experts.

Solution

Out-degree counts escalations sent and identifies agents who ask for help most, which is a workload and training signal. In-degree counts escalations received and identifies the people the floor treats as experts. The two lists barely overlap. Ranking on total degree would put a busy agent who both asks and answers a lot at the top, and bury a quiet specialist who is escalated to constantly and never escalates. Here the direction is the finding.

Common mistakes

✗The highest-degree actor is the most important actor. Degree measures direct connection only. An actor with many ties inside one tight group can matter far less than an actor with few ties that span groups, which is why degree, closeness and betweenness are reported together.
✗Degree can be compared straight across two networks. Raw counts depend on network size and on how the relation was measured. A degree of eight in a 15-person team is near saturation, and a degree of eight in a 2,000-person firm is unremarkable. Normalise before comparing, then check that both studies drew the boundary the same way and measured the same relation, because scaling alone does not make two studies comparable.
✗In a directed network the total degree is enough. Adding ties sent to ties received produces a number in which the two meanings cancel out. Someone who asks 10 people for help and someone asked by 10 people share a total degree of 10 and occupy opposite positions.
✗A high average degree means a well-connected network. Averages hide the distribution. Many real networks have a few very high-degree actors and a long tail of low-degree ones, so the mean describes nobody. Look at the spread as well as the centre.

Revision bullets

•Degree is the count of ties; normalised degree divides by n-1
•Purely local: it sees the neighbourhood, never the wider structure
•Directed networks split into in-degree (prestige) and out-degree (activity)
•Raw degree is not comparable across sizes, and normalising alone does not make it so
•Running example: Ana scores 3 divided by 5, or 0.6; Finn scores 0.2

Quick check

Two employees each have a degree of 6 in the same advice network. One sits inside a single tight project team, the other has one contact in each of six departments. What follows?

A consultancy reports that Team A is better connected than Team B because average degree is 7.2 against 4.1. Team A has 12 members and Team B has 90. The comparison is

Connected topics

KPI Design What SNA Is Actors and Ties Weighted Ties Closeness Betweenness Density

More in Networks

Bridges and Cliques Network Study

Sources

Freeman (1978/79)
Freeman, L. C. "Centrality in Social Networks: Conceptual Clarification." Social Networks, 1(3), 215-239, 1978/79.
Sets out degree, closeness and betweenness as three distinct answers to what centrality means, with the normalisations used here.
Wasserman & Faust (1994), Ch. 5
Wasserman, S., & Faust, K. Social Network Analysis: Methods and Applications. Cambridge University Press, 1994.
Covers degree, in-degree and out-degree and their interpretation on directed relations.

How to cite this page

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