Graph Theory / Networks ... Triadic Closure and Strong/Weak Ties

Question

Okay, so the problem I am working on is this:

Consider the graph in the figure below, in which each edge — except the edge connecting B and C — is labeled as a strong tie (S) or a weak tie (W). According to the theory of strong and weak ties, with the strong triadic closure assumption, how would you expect the edge connecting B and C to be labeled? Give a brief (1-3 sentence) explanation for your answer.

Now, I would assume that the edge connecting B and C would be a weak tie (W). My reasoning being that if the B-C edge were to be a strong tie (S), then the nodes B and C would violate the Strong Triadic Closure Property (node B would now have strong ties to both nodes E and C without there being an E-C edge; likewise, node C would have strong ties to both nodes F and B without there being an F-B edge).

Am I on the right track?

Answer 1

Makes sense to me. You're simply applying the definition:

We say that a node A violates the Strong Triadic Closure Property if it has strong ties to two other nodes B and C, and there is no edge at all (either a strong or weak tie) between B and C. We say that a node A satisfies the Strong Triadic Closure Property if it does not violate it.

Page 54 from D. Easley, J. Kleinberg, Networks, Crowds, and Markets: Reasoning about a Highly Connected World (2010) (URL).

As you say, if $bc$ were a strong tie, then nodes $b$ and $c$ would violate the Strong Triadic Closure Property. And, under this model, there are only strong ties and weak ties (and non-ties), so a tie that is not strong must be weak.