Re: eigenvector centrality

• To: mathgroup at smc.vnet.net
• Subject: [mg101918] Re: eigenvector centrality
• From: antononcube <antononcube at gmail.com>
• Date: Wed, 22 Jul 2009 06:25:04 -0400 (EDT)
• References: <h3n5pr\$2bc\$1@smc.vnet.net>

```> The principle eigenvector of a non-negative matrix is also non-negative, according to the Perron=96Frobenius theorem.

Sure. Let us denote your matrix with A and the pricipal value and
vector with p and v respectively. I.e. we have

A . v == p  * v .

Then this means that the following equation is also true:

A . (-v) == p ( -v)

Note, that the first vector of the computed eigensystem is with all
negative coordinates and hence satisfies the equation above, i.e.

In[785]:= {vals, vecs} =
Eigensystem[
N[{{0, 1, 5, 0, 11, 6, 3, 5, 1, 3, 36, 17}, {1, 0, 1, 1, 0, 7, 0,
0, 1, 2, 0, 0}, {5, 1, 0, 1, 6, 1, 0, 1, 0, 1, 40, 1}, {0, 1, 1,
0, 0, 0, 0, 0, 2, 1, 1, 0}, {11, 0, 6, 0, 0, 6, 1, 0, 3, 0, 9,
0}, {6, 7, 1, 0, 6, 0, 2, 3, 3, 0, 8, 0}, {3, 0, 0, 0, 1, 2, 0,
8, 2, 0, 1, 20}, {5, 0, 1, 0, 0, 3, 8, 0, 2, 1, 2, 24}, {1, 1,
0, 2, 3, 3, 2, 2, 0, 6, 0, 0}, {3, 2, 1, 1, 0, 0, 0, 1, 6, 0, 1,
0}, {36, 0, 40, 1, 9, 8, 1, 2, 0, 1, 0, 0}, {17, 0, 1, 0, 0, 0,
20, 24, 0, 0, 0, 0}}]];

In[786]:= Abs[vecs[[1]]] == -vecs[[1]]
Out[786]= True

--Anton

```

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