# Math Insight

### Image: Degree distribution of a directed network

The degree distribution of a directed network with 10 nodes and 13 links.

The network has adjacency matrix \begin{gather*} A \nobreak{=}\left[ \begin{array}{cccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right]. \end{gather*} Its in- and out-degrees are $(k_1^{\text{in}},k_1^{\text{out}})=(0,2)$, $(k_2^{\text{in}},k_2^{\text{out}})=(2,1)$, $(k_3^{\text{in}},k_3^{\text{out}})=(0,1)$, $(k_4^{\text{in}},k_4^{\text{out}})=(1,0)$, $(k_5^{\text{in}},k_5^{\text{out}})=(2,0)$, $(k_6^{\text{in}},k_6^{\text{out}})=(3,2)$, $(k_7^{\text{in}},k_7^{\text{out}})=(3,3)$, $(k_8^{\text{in}},k_8^{\text{out}})=(0,2)$, $(k_9^{\text{in}},k_9^{\text{out}})=(2,1)$, $(k_{10}^{\text{in}},k_{10}^{\text{out}})=(0,1)$.

Source image file: small_directed_network_degree_distribution.py
Source image type: Python