Network Math

Metcalfe’s law explains why networks are so valuable – it says that the value of a network is proportional to the square of the number of users. When Kevin Kelly explains this, he illustrates it by saying that the first person with a fax machine was an idiot. What can you do with the only fax machine in the world? Once there are two, you can communicate with each other. And as the network grows, according to Metcalfe’s law, the value created by the network grows even faster than the number of people in it.

This is true of fax machines, of telephones, of internet users, and of social networks. Once we get to very large networks, the number of potential connections available to us is staggeringly large. This high level of potential connectivity is important. It’s one of the reasons that people are willing to stick with Facebook despite each new abuse of their private information – everyone is on Facebook. You can switch to a different social network with better privacy policies, but because currently all of the competing networks are much smaller, the potential value to people is much lower as well.

While the potential connectivity creates the value in these networks, there is a potential downside to large networks. We want to have the ability to reach a large number of people, but we can’t maintain connections with everyone. These large networks tend to be pretty sparsely connected, because it is expensive to build and maintain connections. Think about how much time you’d spend if you were friends with everyone on Facebook – your newsfeed would be impossible to keep track of, and you wouldn’t be able to separate out the news from people you actually care about. At a more basic level, while it’s pretty cool that when you have a telephone, you can call everyone in the world, but what if everyone in the world suddenly decided to call you? High levels of connectivity in large networks are overwhelming.

This is why network structure is important. Every real network consists of a set of ties that are actually a small subset of the total number of possible ties. Consequently, the structure of these ties is very important. Here are some examples from Valdis Krebs’ excellent blog The Network Thinker – a group of ten people might be organised like this:

There are 90 possible ties between 10 people (n * (n-1)) – but this network only has 9 ties. The density is 10% – which is fairly typical for an organisational network. You can see that it has a high degree of structure – it is a hierarchy. In fact, it is an org chart. That’s one way that we can visualise how the group is organised and interacts.

However, if you look at who actually works together within this group, you get a very different picture:

That network is still sparsely connected, but you can see that the structure is quite different. Why is this important? Valdis’ original post makes the point beautifully. We need to understand our actual work-flow and information-sharing network structures in order to get the results that we want within our groups.

Org charts rarely reflect the true network structure of a team. Our actual networks are never fully connected. Metcalfe’s law suggests that the value in large networks is in the potential ability to reach everyone. But in practice, you can’t be connected to everyone. Consequently, the actual structure of connections within a network becomes important. It can be highly structured, like a hierarchy. It can be completely chaotic, with nothing more than random connections between members. Or it can look more like the work-flow network above. In any case, we need to understand the actual structure of our networks if we want them to perform well.

Student and teacher of innovation - University of Queensland Business School - links to academic papers, twitter, and so on can be found here.

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