

In [2]:

    
%pylab inline









    



Populating the interactive namespace from numpy and matplotlib



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import networkx as nx
import pandas as pd
import seaborn as sns
import scipy as sp
import scipy.stats as sps



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sns.set_style('white')
sns.set_context('notebook')

Lecture 3

Power laws, Preferential Attachment, and master equations

April 5, 2016

Emergenceo of a Giant Component in Erdos-Renyi Random Graphs



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G = nx.random_graphs.erdos_renyi_graph(100, .1)



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def random_graph_maxc(N=100):
    P = np.arange(0,1,.001)
    graphs = [nx.random_graphs.erdos_renyi_graph(N, p) for p in P]
    c_max = [float(len(next(nx.connected_components(g)))) for g in graphs]

    return pd.Series(c_max, index=P*N)



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maxc_df = pd.DataFrame([random_graph_maxc() for _ in xrange(10)])



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maxc_df.median().plot(logx=True, logy=True)









    Out[51]:





<matplotlib.axes._subplots.AxesSubplot at 0x153d49c90>

Preferential Attachment

$P_r(t+1 \rightarrow n_j) = d_j / \sum_i{}{d_i}$



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