Let's say we have a given undirected network with degree distribution $\lambda(k)$, i.e., we have the probability of $\lambda(k)$ to select a node with degree $k$.
Now I am facing a problem that if we have selected a node with degree $k$, that is, we know the center node we selected has exactly the degree of $k$, what is the degree distribution of his/her neighbors?
Intuitively, I think the neighbors' degree distribution is not the same as the whole networks' degree distribution. In some sense, I think it should be like a conditional probability manner like $\lambda(m|k)$, representing the probability of neighbors having the degree of $m$, given the center node have the degree of $k$.
I have tried to look it up, but I only find some definitions like joint degree distribution, marginal degree distribution and degree correlation. Generally speaking, I think the degree correlation is exactly what I want. However, it doesn't provide the conditional probability as I supposed.
I am now trying to combine the joint degree distribution $\lambda(m,k)$ and degree distribution $\lambda(k)$, in order to obtain the conditional probability that I want. I am not sure if it is feasible.
By the way, are there some existing conclusions about this topic of Erdos-Renyi random networks, Barabasi-Albert scale-free networks, and any graph structure? I would be really appreciated if anyone could give me some introductions or papers!
Best Answer
If I understand your question correctly, what you are looking for is exactly $ Pr(deg(u) = k_u | deg(v) = k_v) $ where $u$ and $v$ are neighbors. Then this is established (for the stationary state) by Fotouhi, Rabbat (2013) [1] and is as follows
$$Pr(deg(u)=l|deg(v)=k) = \frac{\beta (k+2)}{kl(l+1)} - \frac{\beta}{kl}B^{2\beta+2}_{\beta+1} \frac{B^{k+l-2\beta}_{l-\beta}}{B^{k+l+2}_{l}}$$
where $B^i_j$ is the binomial coefficient (i choose j). This is for the Barabasi-Albert network generated through simple preferential attachment with parameter $\beta$ (i.e. the number of edges added for each new node in the process is $\beta$) as $t\rightarrow +\infty$. So, for the finite case, especially for smaller networks, you should normalize it such that the sum of these probabilities over all values of $l$ is $1$ (You can refer to Bertotti, Modanese 2019 [2] for this).
In an Erdos-Reneyi random graph degrees are not correlated (other than the fact that there is an edge between two neighbors). In Bertotti, Modanese (2019) [3] they discuss the configuration model with Newman rewiring, which accounts for degree correlation of neighbors. You may find that paper ([3]) itself and its references helpful for the general case.
References:
[1] Fotouhi, B., & Rabbat, M. G. (2013). Degree correlation in scale-free graphs. The European Physical Journal B, 86(12), 510.
[2] Bertotti, M. L., & Modanese, G. (2019). The Bass diffusion model on finite Barabasi-Albert networks. Complexity, 2019.
[3] Bertotti, M. L., & Modanese, G. (2019). The configuration model for Barabasi-Albert networks. Applied Network Science, 4(1), 32.