- knn_w(1)
- www.complex-networks.net
- knn_w(1)

`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).

`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.

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>`

.

- www.complex-networks.net
- September 2017
- knn_w(1)