Before revision:  For finite graphs, the index depends on the adjacency eigenvalues of the graphs. Using the eigenvalues of the associated graphon operator, we are able to find an approximation of the index that is tight when the network size goes to infinity.
After revision:  For finite graphs, the index depends on the adjacency eigenvalues of its graph. We then assume that the graph is a random sample from a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of the associated graphon operator, we are able to find an approximation of the index that is tight when the network size goes to infinity.