I was wondering how Barabsi-Albert algorithm assigns $m$ new edges to the nodes with different weights (duplicated edges are prohibited). I could merely justify what is the probability for a specific node, like $i$, to be selected: It's said that the probability of choosing that node equals:
$$P(\bigcup_{j=1}^{m}\{\text{edge j attached to i}\})=\sum_{j=1}^{m}p_i=mp_i$$
But, I want to determine the probability space of this assignment. In other words, Let's consider there are $n$ boxes with weights $p_1,\cdots,p_n$, where $\sum_j p_j=1$, and we're going to choose $m\leq n$ ones regarding the weights.
Indeed, I cannot propose a well-defined probability measure for $(\Omega,\mathcal F,P)$, in which $\Omega$ is the set of subsets of order $m$, nor could I find out how this assigning is modeled/simulated--is it done simultaneously or by a process.
The first try is to let, for example, $P\{1,\cdots,m\}:=p_1\cdots p_m$, but it obviously lacks something.