Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal. 


Let $f({\bf X})$ be a (undisclosed) linear transformation of a matrix $X$. Let ${\bf y}$ and ${\bf y'}$ be vectors, and ${\bf 1}$ denote the vector of all ones. 


I'm using the matrix inversion lemma (Woodbury formula) to rewrite 
$${\bf y'} = f({\bf A'})^{-1}f({\bf A}){\bf y}$$
as 
$${\bf y'} = {\bf y} - f({\bf A})^{-1}{\bf Q}\left[{\bf \Lambda^{-1}} + {\bf Q}f({\bf A})^{-1}{\bf Q}\right]^{-1}{\bf Q}^{-1}{\bf y}$$
where I use the eigendecomposition $${\bf P = Q^{-1}\Lambda Q}.$$


Unfortunately, ${\bf A'}$ is unknown, so ${\bf P}$ is a random symmetric matrix that is constrained away from being pure Bernoulli. (Even if it were Bernoulli, it appears most of the random matrix literature focuses on the case of p=0.5, whereas I'm most interested in the case of p << 0.5.)


Is it possible to determine the probability distribution of ${\bf Q}$ and ${\bf \Lambda}$ at this point in the random matrix literature? (I am an outsider in an applied field.)



Distribution of eigenvectors and eigenvalues for random, symmetric matrix
It sounds like the OP has a random perturbation of a fixed graph, which is not considered very frequently, but when they have, it seems to be by A. Flaxman (see, e.g.:


Expansion and lack thereof in randomly perturbed graphs
AD Flaxman - Internet Mathematics, 2007 - Taylor & Francis


)


Eigenvectors are less-well understood. For a comparison of both sorts of questions in a number of models (sadly, none the perturbation model of a fixed graph), see:


*Rivin, Igor*, [**Spectral experiments+**](http://dx.doi.org/10.1080/10586458.2015.1060908), Exp. Math. 25, No. 4, 379-388 (2016). [ZBL1375.60025](https://zbmath.org/?q=an:1375.60025). 


