Consider a network of $N$ nodes with no self loops described by the adjacency matrix $A \in \{0, 1\}^{N \times N}.$ Let's define as $$\deg(v) = \sum_{w=1}^N a_{v,w}$$ the degree of node $v$. Let's assume that $\deg{v} \geq 1$ for all $v$. Suppose that the distribution of degrees is known, i.e.:
$$\mathbb{P}(\deg(v) = k) = p_k,$$
for $k \in \{1, \ldots, N-1\}$. Suppose now that:
- We randomly extract a node $v$ from the network;
- We randomly extract a node $w$ from the neighbourhood of $v$ (i.e. $\{w : a_{v,w} = 1\}$).
How can I calculate
$$\mathbb{P}(\deg(w) = k_w |\deg(v) = k_v)$$
knowing just $p_k$?
Addendum 1 I'm particularly interested into finding this conditional distribution for scale free (preferential attachment) and Erdos-Renyi graphs.
This is used a lot in research related to epidemic spreading on networks in SIS models. To calculate this, you need to know the joint degree distribution which is usually denoted by $e(k,k')$ which gives the probability that an edge chosen uniformly at random has nodes with degrees $k,k'$ at its ends. Then, the quantity you are interested in is exactly $\frac{e(k,k')}{q(k')}$ where $q(k')$ is the marginal of the joint distribution $e(\cdot,\cdot)$. For more information, have a look at:
-- Epidemic spreading in complex networks with degree correlations (Marian Boguna, Romualdo Pastor-Satorras, Alessandro Vespignani)
-- The "majority illusion" in social networks (K Lerman, X Yan, XZ Wu)
Hope this helps.