NetBunch/doc/knn_w.md

knn_w(1) -- Compute the weighted average nearest neighbours degree function

SYNOPSIS

knn_w <graph_in> [<NO|LIN|EXP> <bin_param>]

DESCRIPTION

knn_w computes the weighted average nearest neighbours degree function knn_w(k) of the weighted graph <graph_in> given as input. The program can (optionally) average the results over bins of equal or exponentially increasing width (the latter is also known as logarithmic binning).

PARAMETERS

• <graph_in>: undirected and weighted input graph (edge list). If is equal to - (dash), read the edge list from STDIN.

• NO: If the second (optional) parameter is equal to NO, or omitted, the program will print on output the values of knn_w(k) for all the degrees in <graph_in>.

• LIN: If the second (optional) parameter is equal to LIN, the program will average the values of knn_w(k) over <bin_param> bins of equal length.

• EXP: If the second (optional) parameter is equal to EXP, the progam will average the values of knn_w(k) over bins of exponentially increasing width (also known as 'logarithmic binning', which is odd, since the width of subsequent bins increases exponentially, not logarithmically, but there you go...). In this case, <bin_param> is the exponent of the increase.

• <bin_param>: If the second parameter is equal to LIN, <bin_param> is the number of bins used in the linear binning. If the second parameter is EXP, <bin_param> is the exponent used to determine the width of each bin.

OUTPUT

The output is in the form:

    k1 knn_w(k1)
    k2 knn_w(k2)
    ....

If no binning is selected, k1, k2, etc. are the degrees observed in <graph_in>. If linear or exponential binning is required, then k1, k2, etc. are the right extremes of the corresponding bin.

EXAMPLES

To compute the average neanest-neighbours degree function of the US air transportation network we can run:

      $ knn_w US_airports.net
      1 81.8
      2 30.350938
      3 15.198846
      4 15.046341
      5 13.967998
      6 16.293341
      7 11.746223
      8 11.53912
      9 7.9134643
      10 8.317504
      ....
      132 0.46248989
      136 0.47312661
      145 0.37386548
      $

Since we have not requested a binning, the program will output the value of knn_w(k) for each of the degrees actually observed in the input graph (the mininum degree is 1 and the maximum degree is 145). We can also ask knn_w to bin the results over 10 bins of equal width by running:

    $ knn_w US_airports.net 10
    16 68.359133
    31 89.519255
    46 78.911709
    61 78.802765
    76 76.352358
    91 71.589354
    106 60.433329
    121 62.600988
    136 64.81641
    151 54.210494
    $

or to use instead an exponential binning:

    $ knn_w US_airports.net EXP 1.3
    3 63.062388
    6 70.319368
    10 81.856768
    15 79.766008
    21 96.172011
    29 84.771533
    39 79.591139
    52 80.222237
    69 79.776163
    91 72.217712
    119 61.878435
    155 62.695227
    $

SEE ALSO

knn(1), deg_seq(1)

REFERENCES

• A. Barrat et al. "The architecture of complex weighted networks". P. Natl. Acad. Sci USA 101 (2004), 3747-3752.

• V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Chapter 10, Cambridge University Press (2017)

AUTHORS

(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <v.nicosia@qmul.ac.uk>.