# Average number of edges of an induced graph by using the edge-based node selection technique on a graph with arbitrary degree distribution

Let $$G(U,V,E)$$ be a simple, undirected, bipartite graph and $$U=\{u_1,u_2,{\cdots},u_n\}$$ and $$V=\{ v_{1},v_{2},\cdots,v_{n}\}$$. Let $$d_k^l$$ be the number of vertex with degree $$k$$ in $$l$$, where $$l \in L=\{U,V\}$$ and $$k \in X=\{1,2,\cdots,n\}$$. Note that $$M = |E|= \frac{1}{2}\sum_{i \in X }\sum_{j \in L}d_i^j$$ is the total number of edges of $$G$$.

Now, randomly select $$m$$ edges from $$E$$ without replacement and call a vertex "covered" if it is one of the ends of the randomly selected edges. Then, induce a graph with the covered vertex subset ($$S$$), i.e, a graph $$G_{S}$$ with vertex set $$S$$ and all the edges connecting pairs of vertices in that subset in $$G$$.

The question: is there any way to derive the expected value of the number of edges of $$G_{S}$$ ($$E{[E_{s}]}$$) based on the probability of edge selection $$p_{e} = \frac{m}{M}$$ and the degree distribution of $$G$$?

For the case of the uniform degree distribution, the results in 1 or 2 can be applied, but I would like to see if there is a way to calculate it for an arbitrary degree distribution.