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Is the value of a network really proportional to the square of its size? Or does each new node add limited value?
In July, 2005, Rupert Murdoch's News Corp. paid $580 million for Intermix Media, the owner of MySpace.com, a social network of teens and musicians. In October, 2006, Google bought YouTube.com, a peer-to-peer video clipsharing network of prosumers [1, see references, p. 27] (producers who are also consumers) for $1.65 billion. And, two months later, AT&T paid $86 billion to complete ownership of the wireless and wireline networks of BellSouth and Cingular.
Closing these deals required agreement between buyer and seller regarding fair market value. But what is the best way to value networks such as these? Cash flow? Assets? Revenues? Profitability? Subscribers? Average Revenue Per User (ARPU)?
And, however valued, as a network grows in size, does its value increase? If so, in what proportion relative to its size?
These are important questions, because networks are ubiquitous, whether they are telecommunications networks, transportation networks, neural networks, logistics networks, financial networks or social networks. Valuing them, both statically and accounting for future growth, is important for developing intelligent business strategies and making wise investment decisions.
Connectivity Value
One way to define a network's size is the number of nodes that it has, typically denoted by the variable n. These nodes may be Internet websites, MySpace users, cellular voice and data service subscribers, retail outlets, neurons, etc.
Consider a fully connected network, where every node can communicate or interact with every other node. An example of this would be a plain old telephony network, which allows each customer to call any of the others.
There are a variety of methods for valuing such a network, e.g., the value of collaboration and/or subset-forming ability, as Reed [2] has argued (see "Reed's Law"). But let us focus on the connectivity value of a network, which we define as being the sum, for each pair of nodes, of the value of connectivity between these nodes. For example, for a telephony network comprising three nodes, Alice, Bob and Charlie, the connectivity value of the overall network would be the value Alice and Bob derive from chatting, plus the value that Alice and Charlie get, plus the value that Bob...