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

Weighted ties and tie strength

Q: A recruitment team finds that most successful hires were referred through ties the referrer described as not close. The best explanation is that

weak ties tend to connect otherwise separate groups, so they carry information the close circle does not already have. Close contacts typically know each other and circulate the same openings. A distant contact is more likely to sit in a different pool of information, so a lead arriving through that tie is more likely to be new. The result is structural and does not require the strength ratings to be wrong.

Q: Two analysts compute brokerage on the same email network. One treats every tie as a single hop, the other inverts message counts to get distances. Their rankings differ. Which statement is correct?

They have measured different things: one counts hops, the other counts effort along heavily used channels. Unweighted paths treat a rarely used link as just as cheap to cross as a daily one. Inverting weights makes heavy channels cheap and rare ones expensive, which routes shortest paths differently. Both are valid definitions of distance. The analyst has to say which notion the decision actually needs.

Most real relations vary in intensity, so a tie carries a weight rather than simply existing. Weight might be messages over a seven-day period, dollars transacted, hours worked together or a self-reported closeness rating. Adding up the weights on every tie an actor holds gives its strength, the weighted version of degree. Weight also changes distance, because on a weighted network a strong tie is normally treated as a short one. The strength of weak ties argument says the low-weight connections are often the structurally important ones, since close contacts tend to know each other and hear the same news.

Why it matters

Think of a phone bill. It does not only say who you called, it says for how long. The people you call for an hour a day almost certainly know each other and hear what you hear. The person you call twice a year might be your only route into a completely different circle. Counting contacts treats those two as equal. Weighting them tells you which is which.

Before you read on — recall

A recruitment team finds that most successful hires were referred through ties the referrer described as not close. The best explanation is that

Formulas

Node strength (weighted degree)

s_i = \sum_{j} w_{ij}, \qquad s_i^{\text{in}} = \sum_{j} w_{ji}

On an undirected network the row sum adds the weights on every tie touching actor

i

. On a directed one the row sum is out-strength, the column sum shown here is in-strength, and reporting only one of them hides half the position. Take the six-person team with contact counts over a seven-day period of Ana-Ben 12, Ana-Cara 10, Ana-Dan 2, Ben-Cara 9, Dan-Eve 3, Eve-Finn 8. Ana has strength 24 and Dan has strength 5. Dan is the lightest-connected person by volume and, as the betweenness node shows, still ties Ana at the top on brokerage.

Distance along a weighted tie

d_{ij} = \frac{1}{w_{ij}}

Path algorithms need a cost, and a strong tie should be cheap to cross. Inverting the weight does that, provided every weight is strictly positive and reads as strength, capacity or volume. A tie of weight 10 has length 0.1 and a tie of weight 2 has length 0.5, so closeness and betweenness can then be run on weighted data. If the weight already records a cost, a delay or a dissimilarity, use it directly instead, and never invert a weight of zero or a signed rating.

Tuned weighted degree

C_D^{w}(i) = k_i^{\,1-\alpha} \, s_i^{\,\alpha}

A compromise between counting contacts and summing volume. Setting

\alpha

to zero returns plain degree

k_i

, setting it to one returns strength

s_i

, and values in between trade one against the other. Values above one reward concentration in a few heavy ties.

Worked examples

Scenario

A bank wants to know which relationship managers create the most exposure if they resign. Its customer records log every recorded client contact, so ties can be counted or weighted by contact volume.

Solution

Counting clients puts the manager with 90 small accounts on top. Weighting by revenue or contact hours puts a manager with 11 large accounts on top instead. Neither ranking is wrong, they answer different questions. Revenue at risk follows the weighted view, because the value sits in the heavy ties. Coverage risk follows the unweighted view, because 90 relationships cannot be absorbed by colleagues quickly. Report both, and say which decision each one supports.

Scenario

An analyst has a weighted email network and applies a threshold, keeping only ties with at least 20 messages, so the picture becomes readable.

Solution

The picture is now clean and the conclusion may now be wrong. A threshold removes low-volume ties first, and those are often the ties connecting otherwise separate groups, so the network can fragment into tidy clusters that look like genuine silos. Density falls, brokerage concentrates on a few survivors, and weak bridges can disappear. If a threshold is needed for drawing, draw with it and compute without it, state the cut-off in the write-up, and re-run the analysis at two or three other cut-offs to show whether the conclusion holds.

Common mistakes

✗A stronger tie is always a more valuable tie. Strong ties are often dependable and often redundant, because close contacts frequently know each other and circulate the same information. Weak ties are more likely, though not certain, to reach a different pool, which is why job leads and novel ideas often arrive through them.
✗Frequency of contact measures tie strength. Frequency is one indicator among several and it is easily contaminated by proximity. Two people who sit together interact constantly without being close. Emotional closeness, duration, breadth of topics and mutual favours all carry information that frequency alone misses.
✗Weights can be dropped without consequence because the structure stays the same. Dropping weights keeps the shape and destroys the ranking. An actor with three heavy ties and an actor with three trivial ties become identical, which is precisely the distinction the weights were recorded to preserve.
✗Weights from different sources can simply be added together. Message counts, dollars and survey ratings sit on incompatible scales. Combining them needs an explicit normalisation and a stated rationale, otherwise the largest-scaled source silently dominates every measure computed on the network.

Revision bullets

•Strength is the sum of the weights on every tie touching an actor
•Strong ties are often dependable and often redundant; weak ties more often reach new information
•Weighted path length usually inverts positive weights that record strength
•Thresholding tends to cut weak bridges first, so test several cut-offs
•Frequency is a proxy for tie strength, not a definition of it
•Weights on different scales must be normalised before they are combined

Quick check

A recruitment team finds that most successful hires were referred through ties the referrer described as not close. The best explanation is that

Two analysts compute brokerage on the same email network. One treats every tie as a single hop, the other inverts message counts to get distances. Their rankings differ. Which statement is correct?

Connected topics

Fitness for Use Actors and Ties Degree Centrality Bridges and Cliques Social Data Sources

More in Networks

What SNA Is Closeness Betweenness Density Network Study

Sources

Granovetter (1973)
Granovetter, M. S. "The Strength of Weak Ties." American Journal of Sociology, 78(6), 1360-1380, 1973.
The original argument that weak ties bridge otherwise separate groups and so carry non-redundant information.
Marsden & Campbell (1984)
Marsden, P. V., & Campbell, K. E. "Measuring Tie Strength." Social Forces, 63(2), 482-501, 1984.
Compares candidate indicators of tie strength and finds frequency and duration contaminated by opportunity to meet.
Barrat et al. (2004)
Barrat, A., Barthelemy, M., Pastor-Satorras, R., & Vespignani, A. "The Architecture of Complex Weighted Networks." PNAS, 101(11), 3747-3752, 2004.
Introduces node strength as the weighted counterpart of degree.
Opsahl et al. (2010)
Opsahl, T., Agneessens, F., & Skvoretz, J. "Node Centrality in Weighted Networks: Generalizing Degree and Shortest Paths." Social Networks, 32(3), 245-251, 2010.
Source of the tuning parameter that trades the number of ties against their total volume.

How to cite this page

Dr. Phil's Quant Lab. (2026). Weighted ties and tie strength. Derivatives Atlas. https://phucnguyenvan.com/concept/ba-weighted-ties