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.

Try it yourself

Weighted ties and strength

Ties carry contact counts over a seven-day period. Strength adds the weights an actor holds. For paths a strong tie should be a short one, so distance can be set to 1 divided by the weight, and shortest paths are then found by Dijkstra rather than by counting hops.

Strength of Ana24 contacts

3 ties carrying 12 + 10 + 2 = 24 contacts.

Circle size and shade both show strength, 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 units of 1 divided by the weight, so a heavy tie costs less than one step. Closeness is 5 divided by that sum, which on weighted distance is not capped at one. 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 (1÷w)	Closeness	Harmonic	Betweenness	Clustering
Ana (selected)	3	0.60	24	2.475	2.020	5.249	6	0.33
Ben	2	0.40	21	2.736	1.827	4.953	0	1.00
Cara	2	0.40	19	2.803	1.784	4.537	0	1.00
Dan	2	0.40	5	2.475	2.020	2.113	6	0.00
Eve	2	0.40	11	3.142	1.592	2.872	4	0.00
Finn	1	0.20	8	3.642	1.373	2.626	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 carrying 24 contacts in total. Dropping the weights would leave Ana identical on degree to anyone else holding 3 ties, which is exactly the distinction the weights were recorded to keep. Identical on degree is not identical in position though, because the count says nothing about where those ties lead. Nobody else holds 3 ties at this setting, so there is no one to line up against Ana on that count. The lightest connected actor by volume is Dan on 5 contacts. Dan also sits at the top on brokerage, so volume and position are pulling in opposite directions. Distance is 1 divided by the weight, so Ana-Ben at 12 costs 0.083 to cross while Ana-Dan at 2 costs 0.500. Shortest paths are found by Dijkstra over those costs, which is a different question from counting hops.

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

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. Business Analytics Atlas. https://phucnguyenvan.com/analytics_atlas/concept/ba-weighted-ties