Re: eigenvector centrality

• To: mathgroup at smc.vnet.net
• Subject: [mg101794] Re: [mg101758] eigenvector centrality
• Date: Fri, 17 Jul 2009 05:04:15 -0400 (EDT)
• References: <200907161219.IAA02483@smc.vnet.net>

```Randi,

eigenvectors are defined up to  nonzero scalar multiples: if you multiply by
-1 the first eigenvector (associated to the Perron Frobenius eigenvalue
5.6035) you get the  desired Perron Frobenius eigenvector, which is strictly
positive because your graph is strongly connected.

2009/7/16 Randi Griffin <rgriff23 at gmail.com>

> I have the following adjacency matrix:
>
> {{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}}
>
> and the eigenvector centrality is given by the principle eigenvector of the
> matrix. The principle eigenvector of a non-negative matrix is also
> non-negative, according to the Perron=96Frobenius theorem. So when I do this:
>
> In:
>
> 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}}]]
>
> There should be a non-negative vector corresponding to the largest
> eigenvalue, correct? But there isn't one:
>
> Out:
>
> {{65.6035, -53.3049, 35.2908, -29.1675, -10.336,
>  10.0537, -9.56001, -7.57945, -4.72239, 4.00893, -0.986703,
>  0.700068}, {{-0.503023, -0.0341986, -0.436337, -0.0182901, \
> -0.225812, -0.164597, -0.138333, -0.179206, -0.0405409, -0.0465865, \
> -0.601691, -0.244733}, {-0.48613, 0.0228601, -0.480593, -0.00519367,
>   0.0430584, -0.0431732, -0.0556247, -0.0559529, 0.0105444,
>   0.0225151, 0.690977,
>   0.210115}, {0.0418777, -0.00805576, -0.298485, -0.0137958, \
> -0.0945894, -0.0109749, 0.451137, 0.497785, 0.0463867,
>   0.00827345, -0.281369, 0.60591}, {0.149818, 0.0416835, -0.269249,
>   0.00473048, -0.0384256, -0.144837, 0.345447,
>   0.474367, -0.0393255, -0.0239622,
>   0.192203, -0.705284}, {-0.488027, -0.111176, 0.30536, 0.0339751,
>   0.463177, 0.147206, -0.037637, 0.389012, -0.382326,
>   0.324844, -0.105542, -0.0573198}, {0.0142588, 0.462945, -0.203477,
>   0.152675, 0.227274, 0.489, -0.0229225, -0.0276191, 0.508197,
>   0.379828, -0.120745, -0.10766}, {0.0030468, -0.466958, -0.057771,
>   0.0656396, -0.373669, 0.596064, 0.328831, -0.239789, -0.206541,
>   0.24365, 0.0666447, -0.0853237}, {0.0366873, -0.259941, -0.0393995,
>    0.00577987, -0.295727, 0.301202, -0.647164, 0.515928,
>   0.192504, -0.166889,
>   0.0374135, -0.00307202}, {-0.454972, -0.0600883,
>   0.403505, -0.215574, -0.0754222, 0.118661, 0.347843, 0.0330865,
>   0.519111, -0.399468, 0.0358547, -0.08892}, {-0.0566059, -0.337693,
>   0.169363, 0.318909, -0.187471, -0.474679, 0.000208033, 0.0340935,
>   0.419117, 0.55909, 0.0494938, 0.0073503}, {-0.0215622, 0.274886,
>   0.117357, -0.758944, -0.360433, -0.0578678, -0.0642396,
>   0.0635673, -0.064487, 0.43074, 0.0548435,
>   0.00849165}, {0.189867, -0.539443, -0.268779, -0.496436,
>   0.522222, -0.0197706, -0.0299546, -0.0993414, 0.247183,
>   0.044455, -0.078139, -0.0347375}}}
>
> I can't tell if this is a problem with the way I am using the program or if
> there is something wrong with the way I am going about finding the
> eigenvector centrality of my network.
>
>

```

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