In a network, Metcalfe's value represents the network capacity or the amount of paired connections that can occur. The Metcalfe value would also serve as an upper bound of proportionality.

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The proportionality of Metcalfe's Law is derived from the assertion that from n users, and as more people join the network they increase the value. The value of the network in that case is proportional to that of the square of the users(N²).

Network modeling of Metcalfe's Law

 Peterson, Timothy, Metcalfe's Law as a Model for Bitcoin's Value (January 22, 2018). Alternative Investment Analyst Review, Q2 2018, Vol. 7, No. 2, 9-18., Available at SSRN: https://ssrn.com/abstract=3078248 or http://dx.doi.org/10.2139/ssrn.3078248 

Metcalfe's Law as a Model for Bitcoin's Value

The law was intended to find the factor n where total network costs (cn) are recovered. In some cases, a proportionality factor(A) is applied to the model. It would be represented in a more detailed way as cn= M= A * n(n-1)/2.  Using mathematical reasoning through pair-wise connections, the model assumes equality between the users' network connections.

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