Network Analysis

Brokers, bridges and structural holes, worked on one small network

Weak ties, structural holes and the five broker roles of Gould and Fernandez, computed by hand for a trader who links two villages, and what each measure misses.

Portrait of Stefan NovakStefan Novak · · 10 min read
Brokers, bridges and structural holes, worked on one small network

A trader who buys fish in one village and sells grain in the next, or a go-between who carries a marriage proposal from one family to another, stands where two groups that rarely meet can reach each other through one person. Network analysis has three linked ideas for such positions: the bridge (a property of a tie), the structural hole (a gap between someone's contacts) and the broker (the person across the gap). This piece computes all three by hand on one small hypothetical network, then asks what anthropologists have done with them.

Granovetter's bridges

Mark Granovetter's paper "The strength of weak ties" (1973) starts from tie strength, which he defined as a combination of time spent together, emotional intensity, intimacy (mutual confiding) and reciprocal services. His central assumption concerns triads. If A has strong ties to both B and C, then B and C will almost always be tied as well, at least weakly. He called the triad that breaks this rule forbidden.

A bridge, in the graph theory he borrowed, is a line that provides the only path between two points. If the forbidden triad never occurs, a strong tie can be a bridge only when neither party has any other strong tie, which in a real network hardly ever happens. So bridges are weak ties. Because true bridges are rare in large networks, Granovetter also defined a local bridge of degree n: a tie whose two ends would be n steps apart without it, with n greater than 2. The higher the degree, the more people depend on that shortcut, which is the same work the long-range ties do in the small-world model.

His evidence came from a labor-market study, a random sample of recent professional, technical and managerial job changers living in a Boston suburb. Among the 54 who found their job through a personal contact and were interviewed in person, 16.7 percent saw that contact often at the time (at least twice a week), 55.6 percent occasionally, and 27.8 percent rarely (once a year or less). Useful news about jobs came, more often than not, from the edges of people's networks.

A trader between two villages

Imagine two villages. In the river village, households A, B and C all trade with one another. In the hill village, households D, E and F do the same. T is a trader from the market town who deals regularly with A and B on the river and with D and E in the hills. There are no other ties: seven nodes, ten ties, drawn as two triangles with T between them.

ties who they join
A-B, A-C, B-C river households
D-E, D-F, E-F hill households
T-A, T-B, T-D, T-E the trader and four households

Notice that there is no bridge anywhere in this network. Remove T-A and A still reaches T in two steps, through B, so T-A is not even a local bridge. Yet every route from the river to the hills runs through T. Bridges belong to ties, brokerage to people, and a person can hold a gap open with redundant ties on each side of it.

Betweenness: how much traffic passes the trader

Betweenness counts the shortest paths between other pairs that pass through a node, with fractional credit when several shortest paths exist (the explainer on density and centrality works through the rules). Seven nodes make 21 pairs.

  • The 9 pairs that join a river household to a hill household all pass through T, and nobody else can carry them. T scores 9.
  • C reaches T by two equally short routes, through A or through B, so A and B each earn 1/2 for the pair C-T. The same split applies to C-D, C-E and C-F, since each of their shortest paths starts C-A-T or C-B-T. A therefore earns 4 × 1/2 = 2, and so does B. By symmetry D and E earn 2 each.
  • C and F sit at the far corners and earn nothing.

A check: the scores add to 9 + 2 + 2 + 2 + 2 = 17, which must equal the sum over all 21 pairs of (distance - 1). The ten tied pairs add 0; six pairs at distance 2 (T-C, T-F, A-D, A-E, B-D, B-E) add 1 each; four at distance 3 (A-F, B-F, C-D, C-E) add 2 each; C-F, at distance 4, adds 3. That is 6 + 8 + 3 = 17. The largest score one node could reach here is the number of pairs that leave it out, 6 × 5 / 2 = 15, so T's normalized betweenness is 9/15 = 0.60.

Burt's structural holes

Ronald Burt's Structural Holes (1992) moved attention from the tie to the gap. A structural hole is the absence of a tie between two of your contacts. Contacts who know each other hear the same news, so ties to both are partly redundant; contacts on opposite sides of a hole bring different information, and whoever sits between them can decide what passes. Two of Burt's indices are easy to compute by hand.

Effective size is the number of contacts minus their redundancy. For a network without tie weights, Stephen Borgatti (1997) showed that it reduces to

effective size = n - 2t / n

where n is the number of ego's contacts and t the number of ties among those contacts. Constraint asks how much of ego's network leads back to each contact, directly or through mutual contacts. Write p(i,j) for the share of i's ties that go to j, which in an unweighted network is 1 divided by i's degree. The constraint that contact j puts on ego i is

c(i,j) = (p(i,j) + Σ p(i,q) × p(q,j))^2

where the sum runs over the contacts q that i and j share. Ego's total constraint is the sum over all of ego's contacts.

Start with T. It has n = 4 contacts (A, B, D, E) and t = 2 ties among them (A-B and D-E). Effective size is 4 - 2 × 2 / 4 = 3. For contact A, T's direct share is 1/4. The only shared contact is B, which adds p(T,B) × p(B,A) = 1/4 × 1/3 = 1/12, because B has three ties. So c(T,A) = (1/4 + 1/12)^2 = (1/3)^2 = 1/9. B, D and E are symmetric with A, so T's total constraint is 4 × 1/9 = 4/9 ≈ 0.44.

Now A, whose contacts B, C and T have two ties among them (B-C and B-T): effective size 3 - 4/3 ≈ 1.67. For contact B the direct share 1/3 is joined by 1/3 × 1/2 through C (two ties) and 1/3 × 1/4 through T, so c(A,B) = (7/12)^2 = 49/144 ≈ 0.34. Contacts C and T each give (1/3 + 1/9)^2 = 16/81 ≈ 0.20. A's total is about 0.74 (exactly 953/1296).

C has two contacts who know each other. Its effective size is 2 - 2 × 1 / 2 = 1, and its constraint is 2 × (1/2 + 1/2 × 1/3)^2 = 2 × 4/9 ≈ 0.89.

node contacts ties among contacts effective size constraint betweenness
T 4 2 3.00 0.44 9
A 3 2 1.67 0.74 2
C 2 1 1.00 0.89 0

The trader has the most non-redundant contacts and the least constraint; corner household C has the reverse. Studying managers in a large American electronics company, Burt (2004) found that people whose networks spanned structural holes were more likely to voice ideas and have them judged valuable, and fared better on pay, evaluations and promotion.

Five ways to stand in the middle

Betweenness and constraint ignore which groups people belong to. Roger Gould and Roberto Fernandez (1989) built their measure on exactly that. They look at every two-step chain a → b → c in which a has no direct tie to c, so that b is the go-between, and they sort each chain by group membership.

role of b groups in the chain a → b → c plain reading
coordinator all three in one group mediates inside its own group
itinerant a and c share a group, b is outside it an outsider serving one group
representative a and b share a group, c is outside carries its group's business out
gatekeeper b and c share a group, a is outside decides what comes in
liaison three different groups links two groups and belongs to neither

Count T's chains, reading each trading tie as running both ways. Chains inside a village do not count, because A and B already trade directly, and so do D and E. Every chain from a river household to a hill household does count: A → T → D, A → T → E, B → T → D, B → T → E and the four reverse chains, 8 in all. Since T belongs to the market town, all 8 are liaison chains.

If T instead lived in the river village, the structure and betweenness would be unchanged, but the 8 chains would split into 4 representative chains (river to hills) and 4 gatekeeper chains (hills to river). Gould and Fernandez treated the five roles as distinct in kind, not shades of one score, and paired their counts with expected values under random ties, so that a role could be tested rather than asserted.

One weak tie across the gap

Suppose households C and F start trading now and then, through a cousin who moves between the villages. C-F is a local bridge in Granovetter's sense: without it, C and F would be four steps apart (C-A-T-D-F), so its degree is 4. The trader's position shrinks. A and B still reach D and E fastest through T, but the shortest routes from A or B to F, and from C to D or E, now run across the new tie. T's betweenness falls from 9 to 4, while C and F rise from 0 to 2 each. Granovetter's argument expects such a tie to be weak. If C and F grew close, the forbidden triad says that A and B, already close to C, would be drawn into knowing F too, and C-F would stop being the long shortcut it is.

Brokers in the ethnographic record

Anthropology had brokers before it had these measures. Eric Wolf (1956), writing about Mexico, described individuals who mediated between community-oriented groups and groups working through national institutions, and treated them as brokers. Clifford Geertz (1960) applied the idea of a cultural broker to the kijaji, the rural Javanese Islamic teacher standing between village life and the wider religious and national world, and followed how the role was changing as a new national state took shape. Traders and marriage go-betweens fit the same frame, though how any one of them behaves is a question for ethnography, not for the graph.

That is the main caution. Katherine Stovel and Lynette Shaw (2012) stress that brokerage has two faces: it eases exchange between groups that would otherwise stay apart, and it can also breed exploitation, profit-taking and corruption. Structural measures cannot tell the two apart, and they inherit every weakness of the data: if a fieldworker missed the C-F tie, T's betweenness would come out at 9 instead of 4. A high score says where someone sits; it takes observation to learn what they do there. The network density calculator shows how local counts can miss such shifts: paste in the ten ties and it finds four triangles, and adding C-F leaves that count unchanged even though five pairs now have shorter routes.

Sources and further reading

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