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<title>knn_w(1) - Compute the weighted average nearest neighbours degree function</title>
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<a href="#NAME">NAME</a>
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<a href="#SYNOPSIS">SYNOPSIS</a>
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<a href="#DESCRIPTION">DESCRIPTION</a>
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<a href="#PARAMETERS">PARAMETERS</a>
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<a href="#OUTPUT">OUTPUT</a>
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<a href="#EXAMPLES">EXAMPLES</a>
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<a href="#SEE-ALSO">SEE ALSO</a>
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<a href="#REFERENCES">REFERENCES</a>
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<a href="#AUTHORS">AUTHORS</a>
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<ol class='man-decor man-head man head'>
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<li class='tl'>knn_w(1)</li>
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<li class='tc'>www.complex-networks.net</li>
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<li class='tr'>knn_w(1)</li>
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</ol>
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<h2 id="NAME">NAME</h2>
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<p class="man-name">
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<code>knn_w</code> - <span class="man-whatis">Compute the weighted average nearest neighbours degree function</span>
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</p>
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<h2 id="SYNOPSIS">SYNOPSIS</h2>
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<p><code>knn_w</code> <var>graph_in</var> [<var>NO|LIN|EXP</var> <var>bin_param</var>]</p>
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<h2 id="DESCRIPTION">DESCRIPTION</h2>
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<p><code>knn_w</code> computes the weighted average nearest neighbours degree
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function knn_w(k) of the weighted graph <var>graph_in</var> given as input. The
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program can (optionally) average the results over bins of equal or
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exponentially increasing width (the latter is also known as
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logarithmic binning).</p>
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<h2 id="PARAMETERS">PARAMETERS</h2>
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<dl>
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<dt><var>graph_in</var></dt><dd><p> undirected and weighted input graph (edge list). If is equal to
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<code>-</code> (dash), read the edge list from STDIN.</p></dd>
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<dt class="flush">NO</dt><dd><p> If the second (optional) parameter is equal to <code>NO</code>, or omitted,
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the program will print on output the values of knn_w(k) for all the
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degrees in <var>graph_in</var>.</p></dd>
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<dt class="flush">LIN</dt><dd><p> If the second (optional) parameter is equal to <code>LIN</code>, the program
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will average the values of knn_w(k) over <var>bin_param</var> bins of equal
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length.</p></dd>
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<dt class="flush">EXP</dt><dd><p> If the second (optional) parameter is equal to <code>EXP</code>, the progam
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will average the values of knn_w(k) over bins of exponentially
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increasing width (also known as 'logarithmic binning', which is
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odd, since the width of subsequent bins increases exponentially,
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not logarithmically, but there you go...). In this case,
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<var>bin_param</var> is the exponent of the increase.</p></dd>
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<dt><var>bin_param</var></dt><dd><p> If the second parameter is equal to <code>LIN</code>, <var>bin_param</var> is the
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number of bins used in the linear binning. If the second parameter
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is <code>EXP</code>, <var>bin_param</var> is the exponent used to determine the width
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of each bin.</p></dd>
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</dl>
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<h2 id="OUTPUT">OUTPUT</h2>
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<p>The output is in the form:</p>
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<pre><code> k1 knn_w(k1)
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k2 knn_w(k2)
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....
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</code></pre>
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<p>If no binning is selected, <code>k1</code>, <code>k2</code>, etc. are the degrees observed
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in <var>graph_in</var>. If linear or exponential binning is required, then
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<code>k1</code>, <code>k2</code>, etc. are the right extremes of the corresponding bin.</p>
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<h2 id="EXAMPLES">EXAMPLES</h2>
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<p>To compute the average neanest-neighbours degree function of the US
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air transportation network we can run:</p>
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<pre><code> $ knn_w US_airports.net
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1 81.8
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2 30.350938
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3 15.198846
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4 15.046341
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5 13.967998
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6 16.293341
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7 11.746223
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8 11.53912
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9 7.9134643
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10 8.317504
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....
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132 0.46248989
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136 0.47312661
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145 0.37386548
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$
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</code></pre>
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<p>Since we have not requested a binning, the program will output the
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value of knn_w(k) for each of the degrees actually observed in the
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input graph (the mininum degree is 1 and the maximum degree is
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145). We can also ask <code>knn_w</code> to bin the results over 10 bins of equal
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width by running:</p>
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<pre><code> $ knn_w US_airports.net 10
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16 68.359133
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31 89.519255
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46 78.911709
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61 78.802765
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76 76.352358
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91 71.589354
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106 60.433329
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121 62.600988
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136 64.81641
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151 54.210494
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$
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</code></pre>
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<p>or to use instead an exponential binning:</p>
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<pre><code> $ knn_w US_airports.net EXP 1.3
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3 63.062388
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6 70.319368
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10 81.856768
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15 79.766008
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21 96.172011
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29 84.771533
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39 79.591139
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52 80.222237
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69 79.776163
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91 72.217712
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119 61.878435
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155 62.695227
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$
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</code></pre>
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<h2 id="SEE-ALSO">SEE ALSO</h2>
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<p><span class="man-ref">knn<span class="s">(1)</span></span>, <span class="man-ref">deg_seq<span class="s">(1)</span></span></p>
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<h2 id="REFERENCES">REFERENCES</h2>
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<ul>
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<li><p>A. Barrat et al. "The architecture of complex weighted
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networks". P. Natl. Acad. Sci USA 101 (2004), 3747-3752.</p></li>
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<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
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Methods and Applications", Chapter 10, Cambridge University Press
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(2017)</p></li>
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</ul>
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<h2 id="AUTHORS">AUTHORS</h2>
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<p>(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <code><v.nicosia@qmul.ac.uk></code>.</p>
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<ol class='man-decor man-foot man foot'>
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<li class='tl'>www.complex-networks.net</li>
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<li class='tc'>September 2017</li>
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<li class='tr'>knn_w(1)</li>
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</ol>
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