1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
|
import numpy as np
from numpy.linalg import norm
import numpy.random as nr
from scipy.optimize import minimize
import matplotlib.pyplot as plt
import seaborn
from random import random, randint
seaborn.set_style("white")
def likelihood(p, x, y):
a = np.dot(x, p)
return np.log(1. - np.exp(-a[y])).sum() - a[~y].sum()
def likelihood_gradient(p, x, y):
a = np.dot(x, p)
l = np.log(1. - np.exp(-a[y])).sum() - a[~y].sum()
g1 = 1. / (np.exp(a[y]) - 1.)
g = (x[y] * g1[:, np.newaxis]).sum(0) - x[~y].sum(0)
return l, g
def test_gradient(x, y):
eps = 1e-10
for i in xrange(x.shape[1]):
p = 0.5 * np.ones(x.shape[1])
a = np.dot(x, p)
g1 = 1. / (np.exp(a[y]) - 1.)
g = (x[y] * g1[:, np.newaxis]).sum(0) - x[~y].sum(0)
p[i] += eps
f1 = likelihood(p, x, y)
p[i] -= 2 * eps
f2 = likelihood(p, x, y)
print g[i], (f1 - f2) / (2 * eps)
def infer(x, y):
def f(p):
l, g = likelihood_gradient(p, x, y)
return -l, -g
x0 = np.ones(x.shape[1])
bounds = [(1e-10, None)] * x.shape[1]
return minimize(f, x0, jac=True, bounds=bounds, method="L-BFGS-B").x
def bootstrap(x, y, n_iter=100):
rval = np.zeros((n_iter, x.shape[1]))
for i in xrange(n_iter):
indices = np.random.choice(len(y), replace=False, size=int(len(y)*.9))
rval[i] = infer(x[indices], y[indices])
return rval
def confidence_interval(counts, bins):
k = 0
while np.sum(counts[len(counts)/2-k:len(counts)/2+k]) <= .95*np.sum(counts):
k += 1
return bins[len(bins)/2-k], bins[len(bins)/2+k]
def build_matrix(cascades, node):
def aux(cascade, node):
xlist, slist = zip(*cascade)
indices = [i for i, s in enumerate(slist) if s[node] and i >= 1]
if indices:
x = np.vstack(xlist[i-1] for i in indices)
y = np.array([xlist[i][node] for i in indices])
return x, y
else:
return None
pairs = (aux(cascade, node) for cascade in cascades)
xs, ys = zip(*(pair for pair in pairs if pair))
x = np.vstack(xs)
y = np.concatenate(ys)
return x, y
def build_cascade_list(cascades):
x, s = [], []
for cascade in cascades:
xlist, slist = zip(*cascade)
x.append(xlist)
s.append(slist)
return x, s
def simulate_cascade(x, graph):
"""
Simulate an IC cascade given a graph and initial state.
For each time step we yield:
- susc: the nodes susceptible at the beginning of this time step
- x: the subset of susc who became infected
"""
susc = np.ones(graph.shape[0], dtype=bool) # t=0, everyone is susceptible
yield x, susc
while np.any(x):
susc = susc ^ x # nodes infected at previous step are now inactive
if not np.any(susc):
break
x = 1 - np.exp(-np.dot(graph.T, x))
y = nr.random(x.shape[0])
x = (x >= y) & susc
yield x, susc
def uniform_source(graph, *args, **kwargs):
x0 = np.zeros(graph.shape[0], dtype=bool)
x0[nr.randint(0, graph.shape[0])] = True
return x0
def simulate_cascades(n, graph, source=uniform_source):
for t in xrange(n):
x0 = source(graph, t)
yield simulate_cascade(x0, graph)
if __name__ == "__main__":
# g = np.array([[0, 1, 1, 0], [1, 0, 0, 1], [1, 0, 0, 1], [0, 1, 1, 0]])
g = np.array([[0, 0, 1], [0, 0, 0.5], [0, 0, 0]])
p = 0.5
g = np.log(1. / (1 - p * g))
# error = []
def source(graph, t):
x0 = np.zeros(graph.shape[0], dtype=bool)
a = randint(0, 1)
x0[a] = True
if random() > t:
x0[1-a] = True
return x0
thresh = np.arange(0., 1.1, step=0.2)
sizes = np.arange(10, 100, step=10)
nsimul = 10
r = np.zeros(len(sizes), len(thresh))
for t in thresh:
for i in nsimul:
cascades = simulate_cascades(np.max(sizes), g,
source=lambda graph: source(graph, t))
e = np.zeros(g.shape[0])
for j, s in enumerate(sizes):
x, y = build_matrix(cascades, 2)
e += infer(x[:s], y[:s])
for i, t in enumerate(thresh):
plt.plot(sizes, m[:, i], label=str(t))
plt.legend()
plt.show()
# conf = bootstrap(x, y, n_iter=100)
# estimand = np.linalg.norm(np.delete(conf - g[0], 0, axis=1), axis=1)
# error.append(confidence_interval(*np.histogram(estimand, bins=50)))
# plt.semilogx(sizes, error)
# plt.show()
|
