The oscillations in a signal lying on a network can also be quantified by the number of zero crossings in the signal values. If there is an edge between nodes i and j and there is a change in the sign of the signal values at nodes i and j, then it is counted as one zero crossing. Create an undirected random network with 300 nodes by adding an edge between any two nodes with a probability of 0.15.
(a) Plot the number of zero crossings in the eigenvectors of the Laplacian matrix with re-
spect to the corresponding eigenvalues. Comment on your observations and explain the
frequency ordering.
(b) Repeat part (a) for the normalized Laplacian matrix of the network.
The number of zero crossings in the eigenvectors of the Laplacian matrix of a 300 node ER
random network are plotted in Figure 9.3. We can observe that the number of zero crossings
in the eigenvectors of the Laplacian matrix increase with the increase in corresponding eigen-
values. Therefore, the eigenvectors corresponding to small eigenvalues act as low frequency
graph harmonics and vice versa.
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