A Short discussion on Metcalfe's Law for Social Networks

Reading a number of blogs in the last few days, it became clear to a number of us chatting this morning that in fact quite a few Social Media commentators have only a limited knowledge of the information theory behind Social Networking, and that it may be useful to look at some of the theory to understand some of the issues. (Apologies if this sounds a bit lectur-ey, its the frustration factor).

Here is a quick guide to Metcalfe's Law:

Metcalfe's Law states that (From Wikipedia)

the value of a network is proportional to the square of the number of users of the system (n²). First formulated by Robert Metcalfe in regard to Ethernet, Metcalfe's law explains many of the network effects of communication technologies and networks such as the Internet, social networking, and the World Wide Web. It is related to the fact that the number of unique connections in a network of a number of nodes (n) can be expressed mathematically as n(n-1)/2.

Note - Reed's Law (a variant of Metcalfe's Law) says that (Wikipedia again) the utility of large networks, particularly social networks, can scale exponentially with the size of the network. Sarnoff's law (another variant) states that the value of a broadcast network is proportional to the number of viewers.

In fact, we find it more useful to count every connection between 2 users (nodes) as having 2 links, to model a me-to-you link, and a you-to-me link, because they may be asymmetric and have different values (eg bandwidth usage in XDSL systems), ie the formula is n(n-1). This is what is graphed below.



The red line shows utility, i.e. as the number of nodes increases linearly the number of links increases geometrically, increasing the the value of the connections.

So far, so wonderful.

However, what is usually left out is that the capacity of each node to process all this wonderful utility is limited, so at some point the utility hits a barrier where its capacity is 100% consumed - in the graph above it assumes that every new link takes 1% - just 1% - of the utility's transaction capacity, so it hits 100% capacity at just over 10 nodes - 90 links. (This is in effect the theory behind the Dunbar Number, ie the view that a human just cannot manage the transactions required for more than c 150 nodes - thats 22,350 links)

There are ways around this. One way is to lower the per-node Transaction costs, maybe by simplifying the conversation, or by displaying information upfront to reduce explanation, or new technology or whatever. This is the Coasian line above (in honour of Ronald Coase who first discussed the impact of transaction costs in networked structures - firms - in the 1930's - but that is for another post ) and it assumes we can halve transaction costs, and thus double the point of overcapacity to 200%

The other option is to split and define - you don't have to talk to everyone all the time - the third line, the "chain of command" line assumes the structure will split, amoeba like, into subnetworks every time it hits 100% capacity per node. In many human endeavours this is done via a hierarchical chain of command structure.

Incidentally, because each link's communication level varies over time, rather than being a constant 1%, then typically capacity at peak times - when all nodes are communicating more on average - can be less than the theoretical 10 nodes - and unless you want delays to occur, you have to plan for peak capacity in a network - in fact a general rule of any data network is you will hit start to hit peak at roughly 2/3 theoretical maximum capacity - more if the variation of all nodes' comms is high, less if they are lower.

The takeaway here for a Social Network designer?

The transaction costs of a social network will come and bite you at some point - not just with traffic growth (Facebook uses a high transaction cost per interaction between nodes via having to access multiple pages, hence the server issues, we suspect), but also with usefulness - which is why users of Aggregators are increasingly finding that just increasing the number of people communicating , without any increased transaction efficiency (read faster, dammit), or filtering (a form of chain-of-command structure), just overwhelms one with so much data that it becomes noise.

It also shows that although all markets may be conversations, there is a limit to the value of a solely conversational market (high transaction cost per link reduces throughput) so there is a strong incentive to reduce the cost of the conversation by one or both of the transacting parties, especially where the value of the transaction is low

One of the interesting questions we have been kicking around is whether electronic social networks will increase the Dunbar Number, ie the transaction cost is in the logistics of each transaction (meeting and greeting physically) rather than the sheer complexity of managing a large number of transactions - ie is Robert Scoble, with 5,000 friends merely an efficient reducer of tranasction costs, or is each transaction not really a "conversation" as such, but something less?

Comments?

Update - really good post on the risks of taking Metcalfe too literally on Benjamin Ellis's blog (fwiw we covered similar here)