A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions are neighbors if they share a border). Instead, she calculated the row-standardized adjacency matrix and somewhat surprisingly, the entire spectrum was real.


I was wondering whether this was to be expected, or if there is something special about the particular example.


Edit: I have edited out the rest of the question, which was trying to understand the probability. The answer explains why this is always the case.



When does a row standardized adjacency matrix have a real spectrum?
If the adjacency matrix is $A,$ the "row-standardized" matrix is $DA$, where $D$ is a diagonal matrix all of whose diagonal entries are positive, so has a positive diagonal square root $D^{1/2}$. Now, 


$$DA = D^{1/2} D^{1/2} A D^{1/2} D^{-1/2},$$ so your matrix is similar to 


$$D^{1/2} A D^{1/2},$$ which is a symmetric matrix, and so has real spectrum.


