# Math Insight

### Image: Degree distribution of an undirected network

The degree distribution of an undirected network with 10 nodes and 11 links.

The network has adjacency matrix \begin{gather*} A \nobreak{=}\left[ \begin{array}{cccccccccc} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right]. \end{gather*} Its degrees are $k_1=1$, $k_2=3$, $k_3=1$, $k_4=1$, $k_5=2$, $k_6=5$, $k_7=3$, $k_8=3$, $k_9=2$, and $k_{10}=1$ so that its degree distribution is $P_{\text{deg}}(1)=2/5$, $P_{\text{deg}}(2)=1/5$, $P_{\text{deg}}(3)=3/10$, $P_{\text{deg}}(5)=1/10$, and all other $P_{\text{deg}}(k)=0$.

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