I’m using the NetworkX library to work with some small- to medium-sized unweighted, unsigned, directed graphs representing usage of a Web 2.0 site (smallest graph: less than two dozen nodes, largest: a few thousand). One of the things I want to calculate is eigenvector centrality, as follows:
>>> eig = networkx.eigenvector_centrality(my_graph)
>>> eigs = [(v,k) for k,v in eig.iteritems()]
>>> eigs.sort()
>>> eigs.reverse()
However, this gives unexpected results: nodes with 0 outdegree but receiving inward arcs from very central nodes appear at the very back of the list with 0.0 eigenvector centrality (not being a mathematician I may have got this confused, but I don’t think that outward arcs should make any difference to a node’s centrality to a directed graph). In the course of investigating these results, I noticed from the documentation that NetworkX calculates ‘right’ eigenvector centrality by default; out of curiosity, I decided to calculate ‘left’ eigenvector centrality by the recommended method, i.e. reversing the graph before calculating eigenvector centrality (see Networkx documentation). To my surprise, I got exactly the same result: every node was calculated to have exactly the same eigenvector centrality as before. I think this should be a very unlikely outcome (see Wikipedia article), but I have since replicated it with all the graphs I’m working with. Can anyone explain to me what I’m doing wrong?
N.B. Using the NetworkX implementation of the PageRank algorithm provides the results I was expecting, i.e. nodes receiving inward arcs from very central nodes have high centrality even if their outdegree is 0. PageRank is usually considered to be a variant of eigenvector centrality (see Wikipedia article).
Edit: following a request from Aric, I have included some data. This is an anonymised version of my smallest graph. (I couldn’t post toy data in case the problem is specific to the structure of my graphs.) Running the code below on my machine (with Python 2.7) appears to reveal (a) that each node’s right and left eigenvector centrality are the same, and (b) that nodes with outdegree 0 invariably also have eigenvector centrality 0, even if they are quite central to the graph as a whole (e.g. node 61).
import networkx
anon_e_list = [(10, 59), (10, 15), (10, 61), (15, 32), (16, 31), (16, 0), (16, 37), (16, 54), (16, 45), (16, 56), (16, 10), (16, 8), (16, 36), (16, 24), (16, 30), (18, 34), (18, 36), (18, 30), (19, 1), (19, 3), (19, 51), (19, 21), (19, 40), (19, 41), (19, 30), (19, 14), (19, 61), (21, 64), (26, 1), (31, 1), (31, 3), (31, 51), (31, 62), (31, 33), (31, 40), (31, 23), (31, 30), (31, 18), (31, 13), (31, 46), (31, 61), (32, 3), (32, 2), (32, 33), (32, 6), (32, 7), (32, 9), (32, 15), (32, 17), (32, 18), (32, 23), (32, 30), (32, 5), (32, 27), (32, 34), (32, 35), (32, 38), (32, 40), (32, 42), (32, 43), (32, 46), (32, 47), (32, 62), (32, 56), (32, 57), (32, 59), (32, 64), (32, 61), (33, 0), (33, 31), (33, 2), (33, 7), (33, 9), (33, 10), (33, 12), (33, 64), (33, 14), (33, 46), (33, 16), (33, 17), (33, 18), (33, 19), (33, 20), (33, 21), (33, 22), (33, 23), (33, 30), (33, 26), (33, 28), (33, 11), (33, 34), (33, 32), (33, 35), (33, 37), (33, 38), (33, 39), (33, 41), (33, 43), (33, 45), (33, 24), (33, 47), (33, 48), (33, 49), (33, 58), (33, 62), (33, 53), (33, 54), (33, 55), (33, 60), (33, 57), (33, 59), (33, 5), (33, 52), (33, 63), (33, 61), (34, 58), (34, 4), (34, 33), (34, 20), (34, 55), (34, 28), (34, 11), (34, 64), (35, 18), (35, 60), (35, 61), (37, 34), (37, 48), (37, 49), (37, 18), (37, 33), (37, 39), (37, 21), (37, 42), (37, 26), (37, 59), (37, 44), (37, 12), (37, 11), (37, 61), (41, 3), (41, 50), (41, 18), (41, 52), (41, 33), (41, 54), (41, 19), (41, 22), (41, 5), (41, 46), (41, 25), (41, 44), (41, 13), (41, 62), (41, 29), (44, 32), (44, 3), (44, 18), (44, 33), (44, 40), (44, 41), (44, 30), (44, 23), (44, 61), (50, 17), (50, 37), (50, 62), (50, 41), (50, 25), (50, 43), (50, 27), (50, 28), (50, 29), (54, 33), (54, 41), (54, 10), (54, 59), (54, 63), (54, 61), (58, 62), (58, 46), (59, 31), (59, 34), (59, 30), (59, 49), (59, 18), (59, 33), (59, 9), (59, 10), (59, 8), (59, 13), (59, 24), (59, 61), (60, 34), (60, 16), (60, 35), (60, 50), (60, 4), (60, 6), (60, 59), (60, 24), (63, 40), (63, 33), (63, 30), (63, 61), (63, 53)]
my_graph = networkx.DiGraph()
my_graph.add_edges_from(anon_e_list)
r_eig = networkx.eigenvector_centrality(my_graph)
my_graph2 = my_graph.reverse()
l_eig = networkx.eigenvector_centrality(my_graph2)
for nd in my_graph.nodes():
print 'node: {} indegree: {} outdegree: {} right eig: {} left eig: {}'.format(nd,my_graph.in_degree(nd),my_graph.out_degree(nd),r_eig[nd],l_eig[nd])
These two lines
should be replaced with
since the reverse() method by default returns a copy of the graph.