### The Value of Networks

Nothing like a late night cross country flight to think about things like what the equation for a value of a network should be. To summarize from my last post: Metcalf and Reed calculate the theoretical potential value of a network, while its real or realized value should be based on the actual number of links in the network.

So let us a say there is one sub-network on the internet (say, a social networking site), and it has “a” number of people in it. According to Metcalf, the value of the network is a^2. If a second site, say Skype, is added with “b” subscribers, its value would be b^2. So the total value of the internet becomes a^2 + b^2.

To generalize, the total value of the internet should be the sum of its sub networks.

∑(s=1 to n) a(s)^2, where

n is the total number of sub-networks, and

a(n) defines the number of nodes in each of those sub networks.

[Side Note: I hope I got the notation right. It's been years since I last wrote a formula.]

A few insights about this equation:

1. n can be larger than the total number of nodes in the network. So for example if there were N people on the internet, and they all only participated on two social networking sites, then the total value of the network would be 2 * N^2, twice the (simplified) Metcalf value.

2. I think this partially addresses some of the issues Evslin had raised regarding the math of Reed's law.

3. Given a critical mass of population, each incremental node added to a sub network, (a(s)=a(s)+1), can add more value to the total than an addition of a new node to the network (n=n+1). In other words, over time, the size of the internet becomes less important than the networking connections within it.

This last conclusion is important for the implications of openness. Assuming that by opening yourself up (ie allowing your subnetwork to be connected to other sub networks), you increase the reach of your network… then the cost of remaining a closed network C is the difference between the value of a combined network ((a+b)^2) and the value of two separate networks (a^2 + b^2)

C = ((a+b-o)^2) - (a^2 + b^2)

(I have added an adjustment term, "o", to indicate overlap, or any other adjustment…)

Of course, the interesting tension here is that if subnetwork 'a', say Skype, grows its network by itself, it can monetize the full value of the subnetwork, but as pointed out, again by Evslin, if they open up their network, even if they maximize the over all value of the network, they would be decreasing their ability to monetize it. Further more, as a dominant network, they can create the most value by acting like a monopoly. The problem is that businesses can fall into the fallacy of thinking they are Skype, even they don't have the dominance to justify acting like that. And if you don't have the dominant position in a networked market, C can become a problem!

So let us a say there is one sub-network on the internet (say, a social networking site), and it has “a” number of people in it. According to Metcalf, the value of the network is a^2. If a second site, say Skype, is added with “b” subscribers, its value would be b^2. So the total value of the internet becomes a^2 + b^2.

To generalize, the total value of the internet should be the sum of its sub networks.

∑(s=1 to n) a(s)^2, where

n is the total number of sub-networks, and

a(n) defines the number of nodes in each of those sub networks.

[Side Note: I hope I got the notation right. It's been years since I last wrote a formula.]

A few insights about this equation:

1. n can be larger than the total number of nodes in the network. So for example if there were N people on the internet, and they all only participated on two social networking sites, then the total value of the network would be 2 * N^2, twice the (simplified) Metcalf value.

2. I think this partially addresses some of the issues Evslin had raised regarding the math of Reed's law.

3. Given a critical mass of population, each incremental node added to a sub network, (a(s)=a(s)+1), can add more value to the total than an addition of a new node to the network (n=n+1). In other words, over time, the size of the internet becomes less important than the networking connections within it.

This last conclusion is important for the implications of openness. Assuming that by opening yourself up (ie allowing your subnetwork to be connected to other sub networks), you increase the reach of your network… then the cost of remaining a closed network C is the difference between the value of a combined network ((a+b)^2) and the value of two separate networks (a^2 + b^2)

C = ((a+b-o)^2) - (a^2 + b^2)

(I have added an adjustment term, "o", to indicate overlap, or any other adjustment…)

Of course, the interesting tension here is that if subnetwork 'a', say Skype, grows its network by itself, it can monetize the full value of the subnetwork, but as pointed out, again by Evslin, if they open up their network, even if they maximize the over all value of the network, they would be decreasing their ability to monetize it. Further more, as a dominant network, they can create the most value by acting like a monopoly. The problem is that businesses can fall into the fallacy of thinking they are Skype, even they don't have the dominance to justify acting like that. And if you don't have the dominant position in a networked market, C can become a problem!

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