已知 G(n, p) 模型，观察到 m 个边的概率呈：

Given the G(n, p) model, the probability of observing m edges the

Question

已知 G(n, p) 模型，观察到 m 个边的概率呈：

Given the G(n, p) model, the probability of observing m edges the

BlackTom AI · Accepted Answer

In the Erdős–Rényi G(n,p) model each of the N = C(n,2) = n(n−1)/2 possible unordered vertex pairs is an independent Bernoulli trial that yields an edge with probability p. Therefore the total number of edges M is the sum of N independent, identically distributed Bernoulli(p) indicator variables.

Step-by-step reasoning and calculation:
1. Let X_i be the indicator for the i-th potential edge (i = 1,...,N). Then X_i = 1 with probability p and X_i = 0 with probability 1−p.
2. The total number of edges is M = Σ_{i=1}^N X_i.
3. The sum of N independent Bernoulli(p) variables follows a binomial distribution: M ~ Binomial(N, p).
4. Hence the exact probability of observing exactly m edges is
   P(M = m) = C(N, m) p^m (1 − p)^{N − m},
   where N = n(n − 1)/2 and C(N, m) = N! / (m!(N − m)!).
5. The mean and variance follow from binomial properties: E[M] = Np and Var(M) = Np(1 − p).

Why this is correct:
- Independence and identical probability p for each potential edge are exactly the defining assumptions of G(n,p). Summing independent Bernoulli trials yields a binomial distribution, so the binomial formula above is exact.

Why the other choices are incorrect:
- Power-law (degree) distribution: this refers to degree sequences in networks generated by preferential-attachment or scale-free models, not to the global edge-count distribution in G(n,p).
- Poisson distribution: while the Binomial can be approximated by a Poisson(λ) when N is large and p is small with λ = Np fixed, the exact distribution in G(n,p) is binomial, not exactly Poisson. (The option also misspells “Poisson”.)
- Distribution of wealth in society: irrelevant to the G(n,p) random graph model.

Conclusion: The number of edges in G(n,p) follows a binomial distribution. This matches the provided correct answer: "二项分布。binomial distribution."

已知 G(n, p) 模型，观察到 m 个边的概率呈： Given the G(n, p) model, the probability of observing m edges the 单项选择题

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