knn(1) - Compute the average nearest neighbours degree function

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

NAME

knn - Compute the average nearest neighbours degree function

SYNOPSIS

knn graph_in [NO|LIN|EXP bin_param]

DESCRIPTION

knn computes the average nearest neighbours degree function knn(k) of the 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 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(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(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(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(k1)
    k2 knn(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 for a given graph we just run:

      $ knn er_1000_5000.net 
      2 10.5
      3 11.333333
      4 10.785714
      5 11.255319
      6 11.336601
      7 11.176292
      8 11.067568
      9 11.093519
      10 10.898438
      11 10.906009
      12 11.031353
      13 10.73938
      14 10.961538
      15 10.730864
      16 10.669118
      17 10.702206
      18 10.527778
      19 11.302632
      20 11.8
      $

Since we have not requested a binning, the program will output the value of knn(k) for each of the degrees actually observed in the graph er_1000_5000.net (the mininum degree is 2 and the maximum degree is 20). Notice that in this case, as expected in a graph without degree-degree correlations, the values of knn(k) are almost independent of k.

We can also ask knn to bin the results over 5 bins of equal width by running:

    $ knn er_1000_5000.net LIN 5
    6 11.249206
    10 11.037634
    14 10.919366
    18 10.68685
    22 11.474138
    $

Let us consider the case of movie_actors.net, i.e. the actors collaboration network. Here we ask knn to compute the average nearest-neighbours degrees using exponential binning:

    $ knn movie_actors.net EXP 1.4
    2 142.56552
    5 129.09559
    9 158.44493
    15 198.77922
    23 205.96538
    34 210.07379
    50 227.57167
    72 235.89857
    102 254.47583
    144 276.572
    202 307.11004
    283 337.83733
    397 370.34222
    556 410.89117
    779 446.66331
    1091 498.73118
    1527 547.31923
    2137 577.87852
    2991 582.6855
    4187 557.44801
    $

Notice that, due to the presence of the second parameter EXP, the program has printed on output knn(k) over bins of exponentially increasing width, using an exponent 1.4. This is useful for plotting with log or semilog axes. In this case, the clear increasing trend of knn(k) indicates the presence of assortative correlations.

REFERENCES

V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Chapter 7, Cambridge University Press (2017)

AUTHORS

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