可视图复杂网络度分布拟合比较研究

摘要: 可视图（Visibility Graph，VG）算法为研究时间序列的动力学特性提供了复杂网络的思想.网络的度分布反映了时间序列的动力学特征.通过自回归随机过程和分数布朗运动两种不同数据，分别构建可视图.对比结果表明，在自回归随机过程中，度分布可以用指数函数刻画；而在分数布朗运动中，度分布用幂律函数刻画更为合适.这一结论不但适用于VG算法，同时也适用于水平可视图（Horizontal Visibility Graph，HVG）算法.
- 可视图 /
- 自回归随机过程 /
- 分数布朗运动 /
- 拟合优度
Abstract: Visibility graph has provided much insight to study the dynamics of time series from the perspective complex network. We construct visibility graphs for time series from both auto-regressive stochastic and fractional Brownian motions. Our results suggest that degree distributions of the resulted complex networks of auto-regressive processes are characterized by exponential forms, while that of fractional Brownian motions obey power-law forms. Our conclusions hold for both the traditional visibility graph and its variant horizontal visibility graph.
- visibility graph /
- autoregressive stochastic process /
- fractional Brownian motion /
- goodness of fit

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图 1 图 1(a)-图 1(d) 半对数坐标下的度累积分布的指数拟合, 虚线表示率参数 $\lambda$ ; 图 1(e)-图 1(h) 双对数坐标下的度累积分布的幂率拟合, 虚线表示幂律指数 $\gamma$

Fig. 1 (a)-(d) Exponential fitting to the cumulative degree distributions in semi-log plot, and dashed lines are slopes for $\lambda $ ; (e)-(h) Similar to (a)-(d) but power law forms in double log plots

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图 2 在HVG情形下结果, 其余信息如图 1标注

Fig. 2 The same as Fig. 1, except for HVG

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图 3 图 3(a)-图 3(b) 半对数坐标下的度分布的指数拟合, 虚线表示率参数 $\lambda$ ; 图 3(e)-图 3(h) 双对数坐标下的度分布的幂率拟合, 虚线表示幂律指数 $\gamma$

Fig. 3 (a)-(d) Exponential fitting to the degree distributions in semi-log plot, and dashed lines are slopes for $\lambda $ ; (e)-(h) Similar to (a)-(d) but power law forms in double-log plots

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图 4 在HVG情形下结果, 其余信息如图 3标注

Fig. 4 The same as Fig.3, except for HVG

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表 1 AR (1) 在VG算法下指数与幂率拟合比较

Tab. 1 Comparison between exponential and power law scaling factors of VG graphs of AR (1)

	φ=0		φ=0.3		φ=0.9		φ=-0.5
	λ	γ	λ	γ	λ	γ	λ	γ
数值	0.13	4.21	0.12	3.98	0.10	4.09	0.12	4.22
R²	0.998	0.957	0.999	0.919	0.986	0.953	0.998	0.959
SSE	0.655	2.575	0.123	9.074	3.829	5.290	0.440	2.458

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表 2 AR (1) 过程在HVG下的拟合比较，其余信息如表 1标注

Tab. 2 The same as Tab. 1, except for HVG of AR (1)

	φ=0		φ=0.3		φ=0.9		φ=-0.5
	λ	γ	λ	γ	λ	γ	λ	γ
数值	0.45	7.61	0.50	6.54	0.76	8.96	0.43	7.37
R²	0.995	0.947	0.993	0.911	0.994	0.955	0.996	0.952
SSE	1.515	1.599	1.839	3.273	0.951	1.227	0.6017	1.199

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表 3 FBM在VG算法下指数与幂率拟合比较

Tab. 3 Comparison between exponential and power law scaling factors VG graphs of FBM

	H=0.3		H=0.5		H=0.7		H=-0.9
	λ	γ	λ	γ	λ	γ	λ	γ
数值	0.07	2.47	0.03	1.97	0.01	1.56	0.004	1.12
R²	0.989	0.990	0.965	0.995	0.939	0.990	0.962	0.986
SSE	0.527	0.121	6.826	0.105	15.520	0.429	11.770	0.452

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表 4 FBM过程在HVG下的拟合比较，其余信息如表 3标注

Tab. 4 The same as Tab.3, except for HVG of FBM

	H=0.3		H=0.5		H=0.7		H=-0.9
	λ	γ	λ	γ	λ	γ	λ	γ
数值	0.61	11.28	0.73	11.91	0.88	13.26	1.00	9.67
R²	0.975	0.980	0.970	0.980	0.948	0.971	0.953	0.979
SSE	4.689	0.210	6.815	0.274	9.658	0.328	6.999	0.143

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