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.

To make this question more precise, given an Erdos-Renyi random graph, or a "random" Voronoi diagram, are there any theorems about the spectrum of the row standardized adjacency matrix? More precisely, are there any asymptotics about how likely it is that the spectrum is real-valued?

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.