[igraph] Eigenvector centrality

[Matthew Walker]

Hi,

I am trying to understand eigenvector centrality, but I'm clearly doing 
something wrong!

I have found a paper [1] that gives two example networks and their 
eigenvector centrality values.  What is interesting about both these 
example networks is that they contain "positive" and "negative" 
connections.  The paper says that with eigenvector centrality and such 
signed networks, it is possible to classify nodes according to the sign 
of their centrality.

[1] Phillip Bonacich.  2007.  "Some unique properties of eigenvector 
centrality".  Social Networks 29.  Pages 555-564.

For the first network (figure 3A in the paper), iGraph produces a 
(scaled) version of the published results.  I'm very happy with that.  
(The results correctly contain both positive and negative signs.)

However, if I trial the other network (figure 3B), iGraph's results all 
have the same (negative) sign.  This is very different from the 
published results where half the vertices have centrality scores that 
are negative while the other half are positive.  I have checked the 
published results and the file that I hand-wrote by executing 
Network>Centrality>Eigenvector in UCINET.  UCINET produces a (scaled) 
version of the published results.

I have attached the two Pajek-formatted network files.  The published 
eigenvector centrality results are:

Node   3A   3B
1   0.366   0.568
2   0.798   1.024
3   1.367   1.276
4   -1.088   -1.276
5   -1.000   -1.024
6   -1.088   -0.568

The code that I used is copied below.  It writes out the centrality 
scores to x_fact and y_fact for display in Pajek.

What have I done wrong?!  Thanks in advance for your assistance!

Matthew



 // Eigenvector 
centrality                                                                          

 igraph_vector_t lEigenvector;
 igraph_integer_t lNumVertices = igraph_vcount(&lGraph);
 igraph_vector_init(&lEigenvector, lNumVertices);

 igraph_arpack_options_t lArpackOptions;
 igraph_arpack_options_init(&lArpackOptions);

 // Get edge 
weights                                                                                

 igraph_vector_t lEdgeWeights;
 igraph_vector_init(&lEdgeWeights, 0);
 igraph_cattribute_EANV(&lGraph, "weight", 
igraph_ess_all(IGRAPH_EDGEORDER_ID), &lEdgeWeights);

 // Calculate eigenvector centrality
 igraph_eigenvector_centrality(&lGraph,
                               &lEigenvector,
                               NULL, // No eigenvalue 
required                                     
                               0,    // Do not scale 
result                                        
                               &lEdgeWeights, // Edges weights are 
provided                        
                               &lArpackOptions);

 // Set attribute values
 igraph_cattribute_VAN_setv(&lGraph, "xfact", &lEigenvector);
 igraph_cattribute_VAN_setv(&lGraph, "yfact", &lEigenvector);

 // Write Pajek-formatted 
file                                                                     
 FILE* lOutFile;
 lOutFile = fopen("eigenvector.net","wb");
 if (!lOutFile) cerr << "Failed to open file for writing." << endl;
 igraph_write_graph_pajek(&lGraph, lOutFile);
 fclose(lOutFile);



*Vertices 6
1 "One" 0.95 0.25
2 "Two" 0.75 0.05
3 "Three" 0.5 0.25
4 "Four" 0.05 0.25
5 "Five" 0.05 0.95
6 "Six" 0.5 0.95
*Edges
1 2 1
2 3 1
3 4 -1
3 6 -1
4 5 1
5 6 1

*Vertices 6
1 "One" 0.05 0.5
2 "Two" 0.2 0.05
3 "Three" 0.8 0.05
4 "Four" 0.95 0.5
5 "Five" 0.8 0.95
6 "Six" 0.2 0.95
*Edges
1 2 1
2 3 1
3 4 -1
4 5 1
5 6 1