So... you've got some interesting property of the network uncovered. Is it real? How can you trust that what you've found didn't arise by random chance?
One useful way of thinking by using generative models of random graphs. By "generative" and "random", we mean that the graph was generated using some statistical random model underlying it. This is essentially in the "Bayesian" tradition of modelling.
The alternative is a non-parametric way of thinking. This is where we don't assume some model that generated the data, and instead performs randomization on the data on hand to calculate statistics. Here, we assume that our randomization procedure is an appropriate random model for the network.
Discrete Distributions:
n * bernoulli trials follows binomial distribution.Continuous Distributions:
We can also make up an arbitrary distribution of our own. :-)
Assumptions of this model:
Under this model, what is the distribution of number of rides taken throughout the ? We can probably solve this analytically, but since we have ultra-cheap computational capabilities at our fingertips, might as well do it the brute-force way.
In [1]:
import networkx as nx
G = nx.read_gpickle('datasets/divvy_2013/divvy_graph.pkl')
total_trips = sum([d['count'] for _,_,d in G.edges(data=True)])
print(total_trips)
759788
In [2]:
float(total_trips) / len(G.nodes()) ** 2
Out[2]:
8.44208888888889
In [3]:
from collections import Counter
import matplotlib.pyplot as plt
%matplotlib inline
num_trips = [d['count'] for _,_,d in G.edges(data=True)]
plt.bar(Counter(num_trips).keys(), Counter(num_trips).values())
Out[3]:
<Container object of 406 artists>
In [4]:
import numpy as np
np.percentile(num_trips, [2.5, 97.5, 100])
Out[4]:
array([ 1.00000000e+00, 1.10000000e+02, 2.11300000e+03])
Computing the interval between the 2.5th to the 97.5th percentile effectively gives you a centered 95% mass of the distribution. Knowing this range can be quite useful in other data analysis cases.
Together, we will implement a random model for the number of trips that are taken between two stations.
In this procedure, we will copy the nodes graph G into a new graph G_random. This can be done by:
G_random = nx.Graph()
G_random.add_nodes_from(G.nodes(data=True))
Following that, we will manually map-reduce this computation.
In [5]:
# Grab all the nodes from G.
G_random = nx.Graph()
G_random.add_nodes_from(G.nodes(data=True))
G_random.nodes(data=True)
Out[5]:
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'longitude': -87.68955600000001,
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'longitude': -87.6355,
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'longitude': -87.65775,
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'longitude': -87.66850699999999,
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'longitude': -87.67564,
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'longitude': -87.623727,
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'longitude': -87.639566,
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{'dpcapacity': 15,
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'longitude': -87.640826,
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{'dpcapacity': 11,
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'longitude': -87.633523,
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'longitude': -87.631253,
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'latitude': 41.926277,
'longitude': -87.630834,
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'longitude': -87.6658,
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'longitude': -87.640015,
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{'dpcapacity': 15,
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{'dpcapacity': 19,
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{'dpcapacity': 15,
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{'dpcapacity': 15,
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{'dpcapacity': 15,
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{'dpcapacity': 11,
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{'dpcapacity': 19,
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{'dpcapacity': 19,
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{'dpcapacity': 15,
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{'dpcapacity': 15,
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{'dpcapacity': 15,
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{'dpcapacity': 19,
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{'dpcapacity': 15,
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{'dpcapacity': 15,
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{'dpcapacity': 15,
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'longitude': -87.673688,
'name': 'Wolcott Ave & Polk St',
'online date': Timestamp('2013-10-12 00:00:00')}),
(343,
{'dpcapacity': 15,
'landmark': 297,
'latitude': 41.928886999999996,
'longitude': -87.658971,
'name': 'Racine Ave & Wrightwood Ave',
'online date': Timestamp('2013-10-24 00:00:00')}),
(344,
{'dpcapacity': 15,
'landmark': 26,
'latitude': 41.968641,
'longitude': -87.676335,
'name': 'Wolcott Ave & Lawrence Ave',
'online date': Timestamp('2013-10-09 00:00:00')}),
(345,
{'dpcapacity': 15,
'landmark': 119,
'latitude': 41.793242,
'longitude': -87.58778199999999,
'name': 'Lake Park Ave & 56th St',
'online date': Timestamp('2013-10-09 00:00:00')}),
(346,
{'dpcapacity': 15,
'landmark': 353,
'latitude': 41.88283,
'longitude': -87.661206,
'name': 'Ada St & Washington Blvd',
'online date': Timestamp('2013-10-10 00:00:00')}),
(347,
{'dpcapacity': 15,
'landmark': 319,
'latitude': 41.950687,
'longitude': -87.6687,
'name': 'Ashland Ave & Grace St',
'online date': Timestamp('2013-10-12 00:00:00')}),
(348,
{'dpcapacity': 15,
'landmark': 96,
'latitude': 41.854015999999994,
'longitude': -87.69544499999999,
'name': 'California Ave & 21st St',
'online date': Timestamp('2013-10-14 00:00:00')}),
(349,
{'dpcapacity': 15,
'landmark': 210,
'latitude': 41.929142999999996,
'longitude': -87.649077,
'name': 'Halsted St & Wrightwood Ave',
'online date': Timestamp('2013-10-28 00:00:00')}),
(350,
{'dpcapacity': 15,
'landmark': 247,
'latitude': 41.895965999999994,
'longitude': -87.66774699999999,
'name': 'Ashland Ave & Chicago Ave',
'online date': Timestamp('2013-10-22 00:00:00')}),
(351,
{'dpcapacity': 15,
'landmark': 440,
'latitude': 41.803038,
'longitude': -87.606615,
'name': 'Cottage Grove Ave & 51st St',
'online date': Timestamp('2013-10-17 00:00:00')})]
Here is an attempt at writing a numpy-vectorized version of the simulation.
Once again, this is inspired by Jake VDP's PyCon talk on using numpy.
The core idea here is as such:
total trips, but with maximum value edges.np.histogram(trip_edges, bins=num_edges+1). This returns two arrays:This here will be much faster than brute-force iteration because we use numpy functions.
Special thanks to Mauris Van Hauwe on Stack Overflow: http://stackoverflow.com/questions/29578781/how-to-vectorize-simulation-of-distributing-coins-in-numpy/29578883#29578883
In [6]:
# Total number of possible edges.
edges = len(G.nodes())**2
# Recall that the total trips was recorded in the variable total_trips
total_trips
# coins = 800000
# slots = 900000
trip_edges = np.random.randint(edges, size=total_trips)
trips_in_each_edge = np.histogram(trip_edges, bins=np.arange(edges+1))[0]
trips_in_each_edge
plt.hist(trips_in_each_edge)
print(np.mean(trips_in_each_edge), np.var(trips_in_each_edge))
8.44208888889 8.44580185877
Think about it...
Given the distribution of trip counts in the randomly re-distributed trips, what can we infer about the popularity of certain routes? Are they likely to have shown up in this random model?
In [ ]:
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Content source: ehongdata/Network-Analysis-Made-Simple
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