Hi everyone! I'm working on a network analysis project and have some questions about how to calculate and interpret eigenvector centralization in signed network. Hoping someone can help me out.
Considering a signed network with symmetric adjacency matrix like this:
0 -0.5 0.85
-0.5 0 -0.43
0.85 -0.43 0
and its eigenvector centralities of each node are {0.63, -0.48, 0.61}
Here are my questions:
1. How can I calculate the eigenvector centralization of such network using Freeman's method (with the maximum centrality comparing to every centralities; Freeman, 1978)? I found most papers use this method only in unsigned network.
2. How to interpret the centralization I calculated? I have tried to use standard deviation and gini coeffienct to calculate the eigenvector centralization. But I realized that is problematic. For example, even change the sign of centrality above to {-0.63, 0.48, -0.61}, I will still receive the same centralization. In other words, I can't differentiate whether the network is positively or negatively influenced in general.
I'll be more than grateful if anyone could give me some instructions to my question!
Best regards!
Dear Jiahao Zhu,
Review the following video and other data:
https://www.youtube.com/watch?v=IIBOT3SjJZE
https://www.youtube.com/watch?v=9vs1zSqd070
https://www.youtube.com/channel/UCDfHkEuoKb_TXYkQY6BkIIg
Dear Dr. Mizrah Len Leonid Mizrah
Thanks for your sharing! I watched these videos. But unfortunately they don't seem to solve my problems because they introduce eigenvector centrality in unsigned network.
Anyway, I'm still grateful for your answer!
Eigenvector centrality is a more sophisticated view of centrality: a person with few connections could have a very high eigenvector centrality if those few connections were to very well-connected others. Eigenvector centrality allows for connections to have a variable value, so that connecting to some vertices has more benefit than connecting to others.
Let x be eigenvector of the largest eigenvalue λ of the non-negative adjacency matrix A of the undirected graph G = (V, E).
The eigenvector centrality of node i is equal to the leading eigenvector xi of (column) stochastic matrix N := AD−1 (whose leading eigenvalue is 1): Nx = x
Consider a particular node i with its neighboring nodes N(i): xi = ∑ j∈N(i) xj = ∑ j Aijxj
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