The spectral distribution refers to the distribution of the eigenvalues of a characteristic matrix of the network. In KONECT, we show the eigenvalues of the adjacency matrix A, of the normalized adjacency matrix N = D−1/2 A D−1/2, and of the Laplacian matrix L = D − A.
All matrices are of size |V| × |V|, where |V| is the number of nodes in the graph.
The adjacency matrix A contains the edge weights of all edges, or 1 for unweighted networks. A contains 0 for all nodes pairs that are unconnected. For networks with multiple edges, A contains the number of edges connecting any two nodes. The matrix A has both positive and negative eigenvalues in the general case.
The matrix D is the degree matrix of the graph. It is diagonal, and contains the degree of the nodes on the diagonal entries. For weighted networks, the entries are the sum of absolute values of adjacent edge weights.
The matrix N = D−1/2 A D−1/2 is the normalized adjacency matrix. Its eigenvalues are restricted to the range [−1, +1].
The Laplacian matrix L = D − A has only nonnegative eigenvalues, i.e., it is positive-semidefinite.
Note that the bars representing the number of eigenvalues equal to zero is cut off in most plots, since the number of such eigenvalues is usually very high.
Cumulative spectral distribution
The cumulative spectral distribution shows the fraction of eigenvalues of a characteristic graph matrix being larger than a given value. The cumulative spectral distribution is computed for the adjacency matrix A, the normalized adjacency matrix N and the Laplacian matrix L. The X axis shows possible values of the eigenvalues, and the Y axis shows the fraction of eigenvalues being equal or larger than that value.
Eigenvalues are computed approximately in bins. Thus, the plots show boxes enclosing the possible tracings of the actual curve.