knn(1) -- Compute the average nearest neighbours degree function ====== ## SYNOPSIS `knn` [ ] ## DESCRIPTION `knn` computes the average nearest neighbours degree function knn(k) of the graph 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 * : 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 . * LIN: If the second (optional) parameter is equal to `LIN`, the program will average the values of knn(k) over 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, is the exponent of the increase. * : If the second parameter is equal to `LIN`, is the number of bins used in the linear binning. If the second parameter is `EXP`, 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 . 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. ## SEE ALSO knn_w(1), deg_seq(1) ## REFERENCES * V\. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, Methods and Applications", Chapter 7, Cambridge University Press (2017) ## AUTHORS (c) Vincenzo 'KatolaZ' Nicosia 2009-2017 ``.