knn_w - Compute the weighted average nearest neighbours degree function
knn_w graph_in [NO|LIN|EXP bin_param]
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).
undirected and weighted input graph (edge list). If is equal to
- (dash), read the edge list from STDIN.
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.
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.
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.
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.
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.
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
$
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)
(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <v.nicosia@qmul.ac.uk>.