Metcalf's law accused of blowing up the dot-com bubble

The usefulness of the network is directly proportional to the square of the number of users of this network. Robert M. Metcalfe, the inventor of Ethernet, said his famous words in the early 80s in relation to local networks, but ten years later, “Metcalfe’s law” became universally known and began to be applied to any network, be it a telephone network. or internet.

In the late 90s. last century, investors and ordinary people believed in this “magic formula” and inflated the well-known dotcom bubble. Now we are seeing Bubble 2.0 - a kind of repetition of that fever due to the spread of broadband Internet access and fashion on Web 2.0. Therefore, the scientific work published by the famous mathematician Andrew Odlyzko with co-authors is very relevant.

Odlyzhko, formerly the head of the departments of mathematics and cryptography at AT & T Labs, explicitly says that Metcalfe's law had the most dangerous effect during the dot-com boom. Then there was a continuous quantitative growth of the Network: the number of users and the number of sites increased. Venture investors, entrepreneurs, engineers and the simplest people were imbued with the law of Metcalf, which was widely known. They were confident that the usefulness of the Web increases exponentially, even if the number of users grows linearly. Due to the general euphoria, dot-com shares also grew. It's all about math.

Creating local networks, Robert Metcalf noticed that with ten users the maximum possible number of connections in the network is 90. If the network grows twice, to 20 users, then the number of possible connections grows four times to 360. Thus, with Linear investment in Internet business returns will grow exponentially ( graph ).
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A group of American and English mathematicians with the participation of Odlyzhko criticize Metcalf's law for its idealism. In practice, it turned out that the law does not work, because not all network nodes will communicate with each other. In fact, according to Odlyzhko, the value of a network of size n varies according to the formula n log (n) .

This formula is derived from Zipf's law, according to which each new element of the system has proportionally lower value. So, the second element will have a value of ½, the third - 1/3, n-th - 1 / n. Zipf's law is perfect not only for the distribution of words in a natural language, for which he was formulated, but also for describing many other phenomena of reality that exhibit the effect of a “long tail”. For example, Zipf's law and the effect of the “long tail” are manifested in the distribution of the popularity of blogs on the Internet, the sale of goods in an online store, etc.

The formula n log (n) is the embodiment of Zipf's law and is very different from the formula of Metcalf. For example, if you take a two-fold increase in the number of users, then Metcalf's law gives an increase in network value four times, and the logarithmic formula only 2.1 times. As you can see, here, too, more than double the growth, but it is much more realistic growth. The difference in both formulas can be visually evaluated on a comparative graph .

For investors, this is a critical difference. After all, they used the law of Metcalf to evaluate the effectiveness of investments. Now this assessment will have to carry out more "pessimistic" methods.

Of course, the n log (n) formula is a very simplified formula, but, according to experts, it gives the most realistic estimate of the increase in network utility. In real networks, such as the Internet, not all potential connections between nodes are involved. Actually, Robert Metcalf himself spoke about this in his time, but his “law” became more popular than the author’s explanations.