Consider a random bipartite graph $G$ with $K$ left nodes and $M$ right nodes. Each of the $KM$ possible edges of the graph is connected with probability $p$ independently.
I'm trying to compute the expected number of left nodes connected to right nodes of degree $1$, as $K$ and $M$ get large. My first attempt was to observe that there are $Mp$ expected connected right nodes to each left node. By linearity, the expected number of singletons is then
$$ P(\text{this left node is singleton}) = Mp \cdot P(\text{right node is singleton}) $$
However, I'm having a hard time computing $P(\text{right node is singleton})$ since the the number of left nodes connected to each right node is a probability function.
Can anyone help me out? Am I going about this problem the wrong way? Could anyone give me a solution to this problem?
Let $u \in K$ and $v \in M$. I'm using $K$ and $M$ to denote the sets of nodes and theirs sizes, ok?
$$P(u \leftrightarrow v, d(v)=1) = P(d(v)=1 |u \leftrightarrow v)P(u \leftrightarrow v)=(1-p)^{K-1}p$$
Now, let $$N = \sum_{u\in K}\sum_{v\in M} 1_{\{u↔v,d(v)=1\}}$$
note that $N$ counts the number of nodes in $K$ which connect to nodes in $M$ with degree one. Taking the expected value and using linearity you get $$\mathbb{E}[N]=KMp(1-p)^{K-1}$$
You can play with the expression above putting $K$ or $M$ as a function of each other. Or you can set $p$ as a function of $K$ and $M$...
Hope this can help...