Question updated
Suppose that I have an $n\times n$ adjacency matrix $\mathbf{A}$ of a simple graph $G$ where the entry $(i,j)$ represent the number of edges between node $i$ and $j$ in $G$. Note that a simple graph is a unweighted, undirected graph containing no self-loops or multiple edges. All these datasets are scale-free networks whose degree distribution follows a power law, at least asymptotically; that is, the fraction $P(k)$ of nodes in the network having $k$ connections to other nodes goes for large values of $k$ as $P(k) \sim ck^{-\gamma}$ where $c$ is a normalization constant and γ is a parameter whose value is typically in the range $2 < \gamma < 3$, although occasionally it may lie outside these bounds. (This definition is from the corresponding article from Wikipedia.)
I'm interested in applying spectral techniques to analyze real-life large graph datasets. For example, these datasets are available at Stanford Large Network Dataset Collection. These datasets may be collaboration networks (who collaborated with whom), protein-protein interaction networks, or friend relations on a social network such as Facebook or Twitter.
When analyzing a certain property of these scale-free networks, my approach would be much simpler if the adjacency matrix representation of graphs has all distinct eigenvalues. My question is, can I assume that all these matrices have distinct eigenvalues? Very few is known about the spectrum of these matrices, but it is conjectured that the eigenvalue distribution also follows a power law for the highest eigenvalues.
Reference
- Eigenvalue distribution in scale free graphs S. Gago, Aplimat-Journal of Applied Mathematics, 2008