- Mark Granovetter defined the notion of weak ties and showed the role they play in social networks.
- Triadic closure. Process by which links form to close the triangle.
- Bridge. Edge between two nodes that connect two components. The deletion of the bridge would place the two nodes in different components.
- Local bridge. Edge where the two endpoints have no neighbors in common. The two nodes, A and B, can't directly perceive that there's other path.
- Novel information in the alternate component that travels throught the local bridge.
- Strong triadic closure property.
- Violations. A has strong ties to two nodes, B and C, but there is no edge between B and C.
- Using strong triadic closure, we can actually now make the link between the structural property, local bridges, and the interpersonal property of weak ties. And we can actually do it by reasoning mathematically about the graph. So unlike, for example, when we were talking about the global friendship network and we were sort of speculating, here we're not going to speculate. Here we're going to actually start with some very concrete definitions-- a graph, it's edges are divided into strong and weak ties. And say all nodes satisfy the strong triadic closure property, which, again, is a mathematical definition. We then would like to mathematically derive a relationship between local bridges and weak ties. And here is that relationship. And we'll phrase it in the form of a claim. Suppose node A satisfies the strong triadic closure property. Other nodes may or may not, but A does. And suppose that it's involved in at least two strong ties. So that's kind of a very mild assumption. It just says A has at least two close friends in the world. Then here's the conclusion. Any local bridge it is involved in must be a weak tie. So here, for example, we've drawn this network again. Now we've labeled the A to F edge actually so that it's a weak tie. So A, in fact, in this picture satisfies strong triadic closure. We can check that because it's three strong ties with the C, D, and E , and they are all connected. So A satisfies strong triadic closure. It's involved in at least two strong ties. And it is involved in a local bridge to B, that is indeed a weak tie. We'd like to argue that must be true in general. If A satisfies strong triadic closure, at least two strong ties, any of its local bridges must be a weak tie. And here's how we're actually going to try proving that mathematically. We're going to argue by contradiction. We're going to say, all right, suppose by way of contradiction that A has a strong tie to B, and this A-B edge is a local bridge. What we'll try to do is get to a contradiction, a contradiction to our assumption that A satisfies strong triadic closure. And through that contradiction, we'll show that this assumption couldn't hold. And in fact, therefore, that this A-B local bridge must be a weak tie. OK. Well so we don't have a lot to go on when we reason by contradiction because we're only assuming very few things. But one thing we're assuming is that A has at least two strong ties. Well we see one strong tie to B. But there must be at least one other lurking out there somewhere. Let's say it's sum node C. So we've drawn here there's a strong tie to node C. OK. The next step, we also know that the A-B edge is a local bridge. So we know that there is no edge between B and C. Why is there no edge? Because if there were an edge between B and C, then A and B would have C as a common friend and it wouldn't be a local bridge. OK. So there's no B-C edge, but there are strong ties. A has strong ties to both B and C with no edge. That violates the strong triadic closure property. And so, in fact, it does contradict our assumption that A satisfied the strong triadic closure problem. And that's the end of the proof. It shows that under our assumptions any local bridge that A is involved in must be a weak tie. So this claim actually establishes at a formal mathematical level the link that we wanted at a more intuitive level. Right? The local interpersonal property of weak ties that this person's an acquaintance, and the structural property of local bridges, that this edge reaches into another social group. It's a source of novel information. Now, again, I have to stress. We phrased this thing very, very starkly to be able to prove a mathematical statement because we said with a strong triadic closure it's a very strong property. But we can also try to learn from it. What is it telling us at a more qualitative level? Roughly, the point behind the proof is saying that local bridges tend to be weak ties because if they were strong ties, then Triadic Closure would operate around them to shortcut that local bridge to give the two ends mutual friends and it would thereby no longer become a local bridge. So those local bridges actually persist over time. They're likely to be weak ties. That's why they remain local bridges. And in turn, they connect us to these people who travel in other circles, who have novel information, who can give us things like new job leads. And so this way in which these weak ties link us to these other parts of the network that, in the end through this kind of structure, turns out to be the surprising strength of weak ties and social networks.
Source:
- Networks, Crowds and Markets. Module 1. Sections 7 - 9.
