  The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$,
consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A
D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum $\lambda = (\lambda_1,\dots,\lambda_{|V|})$, $0 \le \lambda_1,\le
\dots, \le \lambda_{|V|}\le 2$ of $G$ in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation $\widetilde \lambda = (\widetilde \lambda_1,\dots,\widetilde
\lambda_{|V|})$, $0 \le \widetilde \lambda_1,\le \dots, \le \widetilde
\lambda_{|V|}\le 2$ such that $\|\widetilde \lambda - \lambda\|_1 \le \epsilon
|V|$. Our algorithm has query complexity and running time $exp(O(1/\epsilon))$,
independent of the size of the graph, $|V|$. We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
  In addition we study the implications of our algorithm to property testing in
the bounded degree graph model.
