Network density and centrality, worked by hand

Ask who matters in a village and you will get several answers, each defensible. The household with the most friends. The household everyone can reach quickly. The household that sits on the only route between two clusters of neighbors. Network analysis turns each of those answers into a number, and the numbers disagree with one another in useful ways. This piece builds one small hypothetical network and computes the four measures most anthropologists meet first: density, degree, closeness and betweenness. Everything can be checked with a pencil.

Anthropology has an old claim on this vocabulary. J. A. Barnes is usually credited with the first analytic use of "social network", in his 1954 study of Bremnes, a parish on a Norwegian island, where he needed a term for the web of friendship and acquaintance that cut across classes and committees. A few years later Elizabeth Bott, working with twenty London families, asked whether couples whose friends and relatives all knew one another divided household tasks more sharply than couples whose contacts were scattered. The property she called connectedness is what we now call density.

A hamlet of seven households

Imagine seven households, A to G, in a fishing hamlet. A tie means that two households lent each other a net or a boat during the past season. Lending here runs both ways, so the ties are undirected: if A is tied to B, then B is tied to A. Written as a grid (an adjacency matrix), the data look like this. A 1 marks a tie, a 0 marks none, and the diagonal is blank because a household does not lend to itself.

Draw it and you get two triangles, A, B, C on the left and E, F, G on the right, joined by a short chain C to D to E. There are eight ties: A-B, A-C, B-C, C-D, D-E, E-F, E-G and F-G. Graph theorists call the households nodes (or vertices) and the ties edges; sociologists tend to say actors and ties. The words change, the arithmetic does not. A kinship chart is a network of exactly this kind, with people as nodes and marriage and descent as ties, which is one reason the skills in how to draw a kinship diagram carry over.

If the ties had a direction ("A gave fish to B", which B need not return), the grid would stop being symmetric, and every count below would split into an outgoing and an incoming version. We stay with the undirected case.

Density: how much of the possible is there

With n nodes, the number of possible undirected ties is the number of pairs:

pairs = n(n - 1) / 2 = 7 × 6 / 2 = 21

Density is the share of those pairs that are actually tied. With L ties:

density = L / (n(n - 1) / 2) = 2L / (n(n - 1)) = 16 / 42 ≈ 0.38

So 38 percent of the possible lending partnerships exist. For directed ties the count of possible ties doubles, because A to B and B to A are different arcs, and density becomes L / (n(n - 1)). Twelve arcs among the same seven households would give 12/42 ≈ 0.29.

One identity is worth memorizing, because it explains most of what goes wrong when people compare densities. The average degree (ties per node) is 2L/n, so

density = average degree / (n - 1)

In the hamlet the average degree is 16/7 ≈ 2.29, and 2.29/6 ≈ 0.38, as before. Now imagine a town of 700 households in which each household still has about 2.29 lending partners, a plausible ceiling on how many boats anyone borrows. Density falls to 2.29/699 ≈ 0.003. Nobody has become less neighborly; the denominator grew. Comparing the densities of networks of very different sizes mostly compares their sizes.

Degree: counting ties

The degree of a node is its number of ties, which is just its row sum in the grid. A and B have 2, C has 3, D has 2, E has 3, F and G have 2. The degrees add to 16, twice the number of ties, as they must, since every tie has two ends. That check catches most typing errors in real data.

To compare nodes across networks of different sizes, divide by the largest possible degree, n - 1 = 6. C and E score 3/6 = 0.50; everyone else scores 2/6 ≈ 0.33. By degree, C and E are the central households and D is unremarkable.

Closeness: how far away is everyone else

The distance between two nodes is the length of the shortest path between them, counted in ties. A reaches C in one step and D in two (A to C to D). Here is every distance, with each row's total.

Closeness turns the total upside down, so that being near everyone scores high. In its usual normalized form, closeness = (n - 1) / total distance. For D that is 6/10 = 0.60. For C and E, 6/11 ≈ 0.55. For A, B, F and G, 6/15 = 0.40.

D has overtaken C and E. It has fewer ties, but they point in both directions, so no household is more than two steps from D. A request for a spare net that starts at D reaches the whole hamlet fastest. Closeness in this form only works when every node can reach every other; a household with no ties at all would sit at an infinite distance, and analysts then need a workaround, such as adding up the reciprocals of the distances instead.

Betweenness: who sits on the routes

Betweenness asks how often a node lies on the shortest path between two others. For each pair of other nodes, a node earns 1 if it lies on the only shortest path between them, or a fraction if several shortest paths exist and it lies on some of them.

Work through the 21 pairs in groups:

• Pairs inside the left triangle (A-B, A-C, B-C) and inside the right one (E-F, E-G, F-G) are directly tied. Nobody lies between them.
• Pairs that join a triangle to D: A-D and B-D pass through C, while D-F and D-G pass through E. C-D and D-E are direct.
• The nine pairs that cross from {A, B, C} to {E, F, G} all pass through D. The six that start at A or B also pass through C, and the six that end at F or G also pass through E.

Now add. D lies on the 9 crossing pairs, so its betweenness is 9. C lies on A-D, B-D and six crossing pairs: 8. E, by symmetry: 8. A, B, F and G: 0. The most any one node could score is the number of pairs that do not include it, (n - 1)(n - 2) / 2 = 6 × 5 / 2 = 15, so the normalized scores are D 9/15 = 0.60 and C and E 8/15 ≈ 0.53.

The fractional rule matters as soon as routes multiply. Suppose B and D start lending to each other, a ninth tie. Density rises to 18/42 ≈ 0.43. A now has two equally short routes to D, A to B to D and A to C to D, so B and C each take 1/2 for the pair A-D, and another 1/2 each for A-E, A-F and A-G, which continue through D. C's betweenness falls from 8 to 2, B's rises from 0 to 2, and D keeps its 9. One new tie has largely replaced the left triangle's gatekeeper.

The four measures side by side

For the original eight-tie hamlet:

Degree puts C and E first. Closeness and betweenness put D first. That disagreement is the main lesson. In a 1979 paper in Social Networks, Linton Freeman sorted the many centrality indices then in circulation into three families built on degree, closeness and betweenness. They are not three estimates of one quantity. Degree counts activity, closeness measures reach, and betweenness measures how much of the traffic between other people passes a given door, where it could be helped along, slowed down or overheard.

What the numbers can and cannot say

The numbers describe a structure as recorded, and nothing more. Four cautions follow from the hamlet itself.

The definition of a tie decides everything. Lending boats, sharing food, exchanging spouses over generations and passing on gossip would give four different networks for the same seven households, and D could be central in one and peripheral in the others. A centrality score is always a score on a named relation.

Boundaries move results. D's role as broker exists only among these seven households. If C and E both lent to a household in the next hamlet, that outside household would offer a second two-step route between them, and D's credit for every crossing pair would halve, from 9 to 4.5. Drawing the edge of the network is an analytic decision, and it belongs in the write-up.

Measures differ in how badly they suffer from missing data. Suppose the fieldworker missed a single tie, C-E. Density moves only from 8/21 to 9/21. D's betweenness drops from 9 to 0, because every crossing pair now has a direct shortcut. Density and degree forgive small errors; betweenness does not.

Position is not behavior. A high betweenness score says that D lies on the shortest routes, not that D's members know it, profit from it or control anything. Whether a structural broker acts as a social one is a question for ethnography. Bott's connectedness worked the same way: it let her state an idea about families precisely enough to compare cases, and the comparison still had to be made.

There is a statistical caution too. Ties are not independent observations, since a tie between C and D affects the paths of everyone else, so ordinary significance tests give misleading answers on network data. Permutation tests built for networks are the usual remedy; the statistics starter explains the idea.

If you want to try other configurations without redrawing the grid, the network density calculator does the counting. One exercise to do by hand first: delete the tie D-E and recompute. There are now 7 ties, so density is 14/42 = 1/3. The hamlet splits into two pieces, {A, B, C, D} and {E, F, G}, closeness stops being defined in its simple form, and D, the household that topped two of the three centrality columns, now has a betweenness of zero.

Sources and further reading

• Barnes, J. A. 1954. "Class and committees in a Norwegian island parish." Human Relations 7 (1): 39-58.
• Bott, Elizabeth. 1957. Family and Social Network. London: Tavistock Publications.
• Freeman, Linton C. 1979. "Centrality in social networks: conceptual clarification." Social Networks 1 (3): 215-239.
• Wasserman, Stanley, and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press. https://www.cambridge.org/9780521387071
• Hage, Per, and Frank Harary. 1983. Structural Models in Anthropology. Cambridge Studies in Social Anthropology 46. Cambridge: Cambridge University Press.
• Borgatti, Stephen P., Martin G. Everett, and Jeffrey C. Johnson. 2013. Analyzing Social Networks. London: SAGE (second edition 2018).