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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.)

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    $\begingroup$ I don't really understand why you are doing whatever you are doing. The beginning of the question has nothing random, and then suddenly we have a random matrix. Are the graphs random? If so, what model? If you don't tell us the model, the answer to both questions is NO. $\endgroup$ – Igor Rivin Jan 22 '18 at 1:46
  • $\begingroup$ Thanks for being first to respond! Let me first address where randomness comes from: I begin with deterministic case (where A' is known) for exposition, since I know how to solve for y' there. My problem is in transitioning to probabilistic setting, because A' and P are unknown. To summarize: A is known, but A' and P are unknown. Since A is known, I can solve for y. However, my object of interest is distribution of (y' - y). $\endgroup$ – haz Jan 22 '18 at 2:51
  • $\begingroup$ To address the second question, what model, I do not have one. A, the graph I 'start' with, is some deterministic graph. I then perturb the graph using the matrix P to get A'. You can think of P as thus being the random graph of interest. While I'm interested in a combination of cases (edge addition and removal), I'd place priority on the case of edge addition. In this case, P comes from applying a Bernoulli trial on each edge in the complement graph of A. $\endgroup$ – haz Jan 22 '18 at 2:56
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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+, Exp. Math. 25, No. 4, 379-388 (2016). ZBL1375.60025.

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