# Degree distribution of a graph

Given a graph, what is the degree distribution of the same?

Is degree distribution the same as a histogram of the degrees? As in, is the degree distribution a plot of the number of nodes that have a particular degree?

Secondly, is there a probability component to degree distribution?

• The degree distribution $P(k)$ of a network is then defined to be the fraction of nodes in the network with degree $k$. Thus if there are $n$ nodes in total in a network and $n_k$ of them have degree $k$, we have $P(k) = n_k/n$. en.wikipedia.org/wiki/Degree_distribution – hhsaffar Dec 12 '13 at 15:48
• @hhsaffar So it is essentially just the fraction of nodes having a degree k. Similar to a histogram? – Pravesh Parekh Dec 13 '13 at 10:18

The degree distribution of a nonempty finite graph $G$ with vertex set $V(G)$ is the measure $\mu$ on $\mathbb N_0$ defined by $\mu(\{n\})=\#\{x\in V(G)\mid\deg_G(x)=n\}/\#V(G)$ for every $n$ in $\mathbb N_0$.
If $G$ is a random graph, that is, if $G$ is a random variable defined on some probability space $(\Omega,\mathfrak F,P)$ with values in the space of nonempty finite graphs suitably endowed with a sigma-algebra, then $\mu$ defined as before becomes a random distribution on $\mathbb N_0$, defined by $$\mu(\omega)(\{n\})=\#\{x\in V(G(\omega))\mid\deg_{G(\omega)}(x)=n\}/\#V(G(\omega)),$$ for every $n$ in $\mathbb N_0$.
• @PraveshParekh Suppose you want to know what fraction of the nodes in the graph has a degree equal to $n$. $\mu(n)$ gives you the ratio – hhsaffar Dec 12 '13 at 15:44