Here is an example I wanted to use to illustrate the problems of eigenvector centralities in directed networks. The network has 3 nodes and two edges. $$A\rightarrow B\rightarrow C.$$
Without stongly connected component, I expected the eigencentralities to be all zero. Matlab however tells me otherwise. Now I am not sure if I am making a mistake or if this is a numerical issue. All eigenvalues of the adjacency matrix are zero.
The Matlab program I use is
C = [
0 1 0
0 0 1
0 0 0 ];
CT = C' % Taking the inverse to consider incoming edges.
[V D] = eig(CT);
diagD = diag(D);
maxeigenvalue = max(diagD);
[M,I] = max(diagD);
LeftEigencentrality = abs(V(:,I))