My question is exactly what was asked here:
Specifically:
Given [this definition of $Q$] we proceed by writing $s$ as a linear combination of the normalised eigenvectors $u_i$ of $B$ so that $s = \sum_{i=1}^{n} a_i u_i$ with $a_i = u_i^T\cdot s$.
It's not clear to my why the membership vector $s$ (which can contain only 1 and -1) can be written like this.
The poster never came back to write up the solution, and I do not have the 50 reputation required to comment on Math Overflow, so I can't ask them for the solution.
The paper in question can be found here
This is a linear algebra thing. The matrix B (called the modularity matrix in the paper of interest from the linked question) is real symmetric, and a property of real symmetric matrices is that their eigenvectors form an orthonormal basis. The formula you ask about is just decomposing $s $ as a linear combination of these eigen-basis vectors.