Plotting log-binned network degree distributions


Question

I have often encountered and made long-tailed degree distributions/histograms from complex networks like the figures below. They make the heavy end of these tails, well, very heavy and crowded from many observations:

Classic long-tailed degree distribution

However, many publications I read have much cleaner degree distributions that don't have this clumpiness at the end of the distribution and the observations are more evenly-spaced.

!Classic long-tailed degree distribution

How do you make a chart like this using NetworkX and matplotlib?

1
9
5/10/2013 8:53:55 PM

Accepted Answer

Use log binning (see also). Here is code to take a Counter object representing a histogram of degree values and log-bin the distribution to produce a sparser and smoother distribution.

import numpy as np
def drop_zeros(a_list):
    return [i for i in a_list if i>0]

def log_binning(counter_dict,bin_count=35):

    max_x = log10(max(counter_dict.keys()))
    max_y = log10(max(counter_dict.values()))
    max_base = max([max_x,max_y])

    min_x = log10(min(drop_zeros(counter_dict.keys())))

    bins = np.logspace(min_x,max_base,num=bin_count)

    # Based off of: http://stackoverflow.com/questions/6163334/binning-data-in-python-with-scipy-numpy
    bin_means_y = (np.histogram(counter_dict.keys(),bins,weights=counter_dict.values())[0] / np.histogram(counter_dict.keys(),bins)[0])
    bin_means_x = (np.histogram(counter_dict.keys(),bins,weights=counter_dict.keys())[0] / np.histogram(counter_dict.keys(),bins)[0])

    return bin_means_x,bin_means_y

Generating a classic scale-free network in NetworkX and then plotting this:

import networkx as nx
ba_g = nx.barabasi_albert_graph(10000,2)
ba_c = nx.degree_centrality(ba_g)
# To convert normalized degrees to raw degrees
#ba_c = {k:int(v*(len(ba_g)-1)) for k,v in ba_c.iteritems()}
ba_c2 = dict(Counter(ba_c.values()))

ba_x,ba_y = log_binning(ba_c2,50)

plt.xscale('log')
plt.yscale('log')
plt.scatter(ba_x,ba_y,c='r',marker='s',s=50)
plt.scatter(ba_c2.keys(),ba_c2.values(),c='b',marker='x')
plt.xlim((1e-4,1e-1))
plt.ylim((.9,1e4))
plt.xlabel('Connections (normalized)')
plt.ylabel('Frequency')
plt.show()

Produces the following plot showing the overlap between the "raw" distribution in blue and the "binned" distribution in red.

Comparison between raw and log-binned

Thoughts on how to improve this approach or feedback if I've missed something obvious are welcome.

12
5/14/2013 10:01:57 PM

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