Spectral analysis of networks focuses on the eigenvalues of the graph Laplacian. The Laplacian is a matrix representation of a graph, with a close relationship to the Laplacian operator in the diffusion equation. The eigenvalue spectrum of the Laplacian matrix gives a general characterization of the graph and can be related to almost all important graph invariants.
For the FP networks, spectral analyses have been focused on the Laplacian matrices of the projection networks. On technical grounds, the focus has been on the smaller networks. The eigenvalue spectra can be shown as histograms or as cumulative distributions.
The eigenvectors of the graph Laplacian can be used to calculate optimal projections of the vertices into lower dimensional spaces. Of particular importance is the Fiedler vector, the eigenvector associated with the smallest positive eigenvalue. Using the Fiedler vector, we can see the best one-dimensional version of the vertices, i.e., re-ordering the vertices in a manner that reveals community structure. We again use Netzcope to visualize the adjacency matrix for the network with re-ordered vertices, see community structure for the entire network and for a smaller fragment in greater detail.
