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Eigenvector_centrality power method shifted to A+I #1732

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merged 1 commit into from Aug 11, 2015
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Eigenvector_centrality power method shifted to A+I #1732

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17 changes: 10 additions & 7 deletions networkx/algorithms/centrality/eigenvector.py
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Expand Up @@ -69,12 +69,15 @@ def eigenvector_centrality(G, max_iter=100, tol=1.0e-6, nstart=None,

Notes
------
The measure was introduced by [1]_.
The measure was introduced by [1]_ and is discussed in [2]_.

The eigenvector calculation is done by the power iteration method and has
no guarantee of convergence. The iteration will stop after ``max_iter``
iterations or an error tolerance of ``number_of_nodes(G)*tol`` has been
reached.
Eigenvector convergence: The power iteration method is used to compute
the eigenvector and convergence is not guaranteed. Our method stops after
``max_iter`` iterations or when the vector change is below an error
tolerance of ``number_of_nodes(G)*tol``. We actually use (A+I) rather
than the adjacency matrix A because it shifts the spectrum to enable
discerning the correct eigenvector even for networks with multiple
dominant eigenvalues.

For directed graphs this is "left" eigenvector centrality which corresponds
to the in-edges in the graph. For out-edges eigenvector centrality
Expand Down Expand Up @@ -111,8 +114,8 @@ def eigenvector_centrality(G, max_iter=100, tol=1.0e-6, nstart=None,
# make up to max_iter iterations
for i in range(max_iter):
xlast = x
x = dict.fromkeys(xlast, 0)
# do the multiplication y^T = x^T A
x = xlast.copy() # Start with xlast times I to iterate with (A+I)
# do the multiplication y^T = x^T A (left eigenvector)
for n in x:
for nbr in G[n]:
x[nbr] += xlast[n] * G[n][nbr].get(weight, 1)
Expand Down
3 changes: 3 additions & 0 deletions networkx/algorithms/centrality/tests/test_eigenvector_centrality.py
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Expand Up @@ -41,6 +41,9 @@ def test_P3(self):
b=networkx.eigenvector_centrality_numpy(G)
for n in sorted(G):
assert_almost_equal(b[n],b_answer[n],places=4)
b=networkx.eigenvector_centrality(G)
for n in sorted(G):
assert_almost_equal(b[n],b_answer[n],places=4)


def test_P3_unweighted(self):
Expand Down