_____________ states that the value of a network is equal to the square of the number of users connected to it.

Question

BlackTom AI · Accepted Answer

Metcalfe's Law states that the value of a network is proportional to the square of the number of users connected to it.

1) Relevant knowledge points and concepts:
- Network value is often modeled by counting potential connections between users.
- For a network of n users, the number of unique pairwise connections is given by the combination formula C(n,2).
- Historical context: Robert Metcalfe proposed this relationship, now known as Metcalfe’s Law.
- Other well-known network laws: Sarnoff’s Law (value ∝ n), Reed’s Law (value ∝ 2^n), and Nielsen’s Law (bandwidth growth).

2) Complete reasoning process step by step:
- For n users, each user can potentially connect with (n − 1) other users.
- Counting unique connections (undirected pairs) avoids double-counting, so use combinations:
  C(n,2) = n! / [2!(n − 2)!].

3) Calculations with intermediate steps:
- Compute C(n,2): C(n,2) = n(n − 1)/2.
- Expand: n(n − 1)/2 = (n^2 − n)/2.
- For large n, the dominant term is n^2, so the number of possible connections grows on the order of n^2 (quadratically). Thus network value ∝ n^2.

4) Why the correct answer is correct:
- Metcalfe argued that each potential connection adds value, and because the count of potential connections follows n(n − 1)/2 (≈ n^2/2), the network’s value scales roughly with the square of its user base. This is the essence of Metcalfe’s Law.

5) Why other laws/concepts are not the answer:
- Sarnoff’s Law (linear growth ∝ n) understates value by counting only broadcast reach.
- Reed’s Law (2^n) overstates value by counting all possible group combinations.
- Nielsen’s Law concerns bandwidth, not direct valuation by connections.
- None of these match the “square of the number of users” statement.

6) Conclusion:
The statement describes Metcalfe’s Law. This confirms the correct answer: Metcalfe's Law.

_____________ states that the value of a network is equal to the square of the number of users connected to it.简答题

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