Thursday, November 11, 2010

Metcalfe's and Clancy's Laws

I don't have a Twitter account (years of dealing with administrators have left me averse to anything with "twit" in it), so I'll blog this instead. (In any case, I couldn't do this in 140 characters.  As a professor, I can't do anything in 140 words, let alone 140 characters.) Yesterday Laura McLay blogged about OR and the intelligence community, and mentioned in particular J. M. Steele's paper in Management Science titled “Models for Managing Secrets”.  Steele apparently presents an argument supporting the "Tom Clancy Theorem, which states that the time to detect a secret is inversely proportional to the square of the people that know it (stated without proof in the Hunt for Red October)" (quoting Laura).

The people who know the secret form, at least loosely, a social network. That brings me to Metcalfe's Law, which originally stated that the value of a communications network is proportional to the square of the number of connected devices, but which has since been generalized (bastardized?) to refer to humans as well as, or rather than, devices and to cover social networks as well as communications networks. I imagine Robert Metcalfe intended value to mean value to the aggregate set of owners of the attached devices (which in his mind might well have been one organization, since he's a pioneer in local area networks), but in a social networking context we can think of value as either or both of value to the members of the network (e.g., you and your Facebook friends) or value to the owners of the underlying support structure (e.g., the owners of Facebook.com).

So, letting $T$ be the time to detect a secret, $N$ the number of people who know the secret and $V$ the value of the social network to its members (which presumably includes the originator of the secret), we have $$T=\frac{k_1}{N^2}$$ and $$V=k_2\times N^2,$$ from which we infer $$T=\frac{k_1\times k_2}{V}.$$ Since the value of the social network to the "enemy" is inversely proportional to the time required to reveal the secret, we have direct proportionality between the value of the social network to the entity trying to conceal the secret and the value of that same network to the enemies of that entity.

So now I'm off to cancel my Facebook and LinkedIn accounts.  :-)

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