拟合幂律

Question

我正在尝试重新创建文章中的分析：“从时间序列到复杂网络：可见性图”。以下是文章的链接：https://www.pnas.org/doi/10.1073/pnas.0709247105#core-B14。我正在研究的系列是布朗运动 B(t)，这是我用来生成数据的代码：

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
def brownian_motion(t, a=1):
    dt = t[1] - t[0]
    dW = np.random.normal(scale=np.sqrt(dt), size=t.shape)
    W = np.cumsum(dW)
    return a**(1/2) * W

t = np.linspace(0, 100, 100000)

np.random.seed(42)

W = brownian_motion(t)

plt.plot(t, W)
plt.xlabel('t')
plt.ylabel('W(t)')
plt.title('Brownian Motion')
plt.show()

df = pd.DataFrame(W, index=t, columns=['W(t)'])
df.index.name = 't'
df.to_csv('brownian_motion_data.csv')

以下是我希望达到的结果：

import numpy as np
import matplotlib.pyplot as plt
from ts2vg import NaturalVG
import networkx as nx
import pandas as pd
import powerlaw

data = pd.read_csv('brownian_motion_data.csv')
dat = data['W(t)'][:500]
nodes = range(len(dat))
vg = NaturalVG()
vg.build(dat)
edges = vg.edges
G = nx.Graph()
G.add_nodes_from(nodes)
G.add_edges_from(edges)

n = nx.number_of_nodes(G)
degree_freq = nx.degree_histogram(G)
degree_dist = []
for i in degree_freq:
   x = i / n
   degree_dist.append(x)
degrees = range(len(degree_freq))
fit = powerlaw.Fit(np.array(degree_dist) + 10 ** (-6))
alpha = round(fit.power_law.alpha, 3)

plt.figure(figsize=(8, 6))
plt.scatter(degrees, degree_dist, marker='o', color='b', label='Data')
fit.power_law.plot_pdf(color='r', linestyle='--', label=f'Fit (alpha={alpha})')
plt.xlabel('Degree')
plt.ylabel('Frequency')
plt.legend()
plt.show()

degrees_array = np.array(degrees)
degree_dist_array = np.array(degree_dist)
coefficients = np.polyfit(degrees_array, degree_dist_array, 2)
poly_function = np.poly1d(coefficients)
degrees_fit = np.linspace(min(degrees_array), max(degrees_array), 100)
degree_dist_fit = poly_function(degrees_fit)

plt.loglog(degrees_array, degree_dist_array, label='Original Data')
plt.loglog(degrees_fit, degree_dist_fit, label=f'Polynomial Fit', color='red')
plt.xlabel('Degrees')
plt.ylabel('Degree Distribution')
plt.title('Polynomial Fitting Example for Degree Distribution')
plt.legend()
plt.show()

我的两种方法是使用

powerlaw

和

polyfit

函数。由于度数概率有

0

值，结合我的调整，我得到的结果非常令人失望。我期待一些建议，甚至其他替代方法来使幂律拟合与上述文章类似。

Answer 1

我对幂律包不是很熟悉，但浏览了相应的论文后，我认为您的代码中缺少的是识别幂律拟合的正确数据范围（请参阅“识别缩放范围”部分）。

您的数据显然仅部分遵循幂律。

np.polyfit 函数按预期工作，但您的数据根本不遵循二次多项式。