3
$\begingroup$

As a part of research, I am studying the eigenvalues spectrum of adjacency matrices. My adjacency matrices are symmetrical. However, their elements are following multivariate gaussian distribution. That means there is a correlation between the elements of adjacency matrix. My goal is trying to explain the shape of eigenvalue spectrum which is deviating from semi-circle law based upon of the correlation coefficients. For now, I am just using up to two body correlations. Is there anyone familiar or has done any studies over eigenvalue spectrum for simple non-i.i.d matrices? Or you can recommend some references related what I have in my mind.
I wrote a simple article explaining my question on overleaf. If you like to take a look on it. https://www.overleaf.com/read/znkxqcsccrxy

Improve this question
$\endgroup$
4
  • 2
    $\begingroup$ Can you clarify what you mean? The elements of an adjacency matrix are in $\{0,1\}$. They can't have a multivariate gaussian distribution, because multivariate gaussians are continuous. $\endgroup$
    – Robert Israel
    May 13, 2017 at 1:01
  • $\begingroup$ Or is it some sort of weighted random graphs? $\endgroup$
    – Bach
    May 14, 2017 at 10:57
  • $\begingroup$ By the given adjacency matrix, we can have average and variance. Based upon of that, we can reproduce the corresponding gaussian matrix. Lets say with probability "q" the elements of adjacency matrix can be either zero or 1. I created a sample of such matrices and measure its eigenvalue spectrum. Its eigenvalue spectrum is similar to the spectrum of ensemble of matrices with gaussian entries following the same average and variance. $\endgroup$
    – M. Ezzat
    May 14, 2017 at 18:55
  • $\begingroup$ In my case, I have a model which is so called "XIE" model of extreme introverts and extroverts on a bipartite graph and It is not weighted. As Robert mentioned the adjacency matrix is containing zero's and ones. However, we got interested to analyze the spectrum of adjacency matrices of the model. As a result of the model, the elements of adjacency matrix would be correlated. We measure the two and three body correlations between elements and we are looking into an analytical way to explain the spectrum based upon of measured multi body correlations. $\endgroup$
    – M. Ezzat
    May 14, 2017 at 19:12

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.