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问题 1.

2.11 Average Path Length
The average path length $l$ is defined as the average of the shortest paths between all nodes in the network, i.e.,
$$
l=\left\langle d_{i j}\right\rangle=\frac{1}{N(N-1)} \sum_{i \neq j} d_{i j}
$$
If the graph is disconnected, it makes sense to consider the reciprocal of the harmonic mean; this is because the distance between two nodes belonging to separate components is infinite, the reciprocal being 0 .
$$
l=\left\langle\frac{1}{d_{i j}^{-1}}\right\rangle=\left(\frac{1}{N(N-1)} \sum_{i \neq j} \frac{1}{d_{i j}}\right)^{-1}
$$


问题 2.

3.1 Degree Distribution
We shall describe the concept of degree distribution with the help of an example. Consider the case of citation networks. Scientific papers refer to works done earlier on related topics via citations. In a citation network each node represents a scientific paper and a directed edge from node $A$ to $B$ indicates that $A$ has cited $B$. An important thing to note is that citation networks are acyclic in nature.

Alfred Lotka analysed such networks in 1926. Lotka’s Law describes the frequency of publication by authors in any given field. It states that the number of authors making $n$ contributions to that field is approximately $n^{-\alpha}$ of those making 1 contribution, where $\alpha \approx 2$. This distribution is nothing but the distribution of the degrees of the nodes in the network.

A famous outcome of Lotka’s study was the $80-20$ Rule, which states that $80 \%$ of the people in such a network are $20 \%$ popular, and vice versa.

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