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Reconstruction of dynamic networks with time-delayed interactions in presence of fast-varying noises
| Zhaoyang Zhang | Yang Chen | Yuanyuan Mi | Gang Hu |
|---|---|---|---|
| Ningbo University | 中科院脑网中心和国家模式识别实验室 | Chongqing University | Beijing Normal University |
Dated: April 1, 2019
目录
- Reconstruction of dynamic networks with time-delayed interactions in presence of fast-varying noises
- 理论推导
- 数值仿真
- 总结
理论推导
考虑 N N N个节点的动力学系统
x ˙ i = F i [ x i ( t ) ] + ∑ j = 1 , j ≠ i N Φ i j [ x i ( t ) , x j ( t − τ i j ) ] + η i ( t ) + Γ i ( t ) , i = 1 , 2 , . . . , N \dot{x}_i=F_i[x_i(t)]+\sum_{j=1,j\neq i}^N\Phi_{ij}[x_i(t),x_j(t-\tau_{ij})]+\eta_i(t)+\Gamma_i(t),i=1,2,...,N x˙i=Fi[xi(t)]+j=1,j̸=i∑NΦij[xi(t),xj(t−τij)]+ηi(t)+Γi(t),i=1,2,...,N
其中 η i ( t ) \eta_i(t) ηi(t)为色噪声， Γ i ( t ) \Gamma_i(t) Γi(t)为白噪声，满足
< η i ( t ) > = 0 , < η i ( t ) η i ( t + t ′ ) > = P i j e − ∣ t ′ ∣ τ C <\eta_i(t)>=0,<\eta_i(t)\eta_i(t+t')>=P_{ij}e^{-\frac{|t'|}{\tau_C}} <ηi(t)>=0,<ηi(t)ηi(t+t′)>=Pije−τC∣t′∣
< Γ i ( t ) > = 0 , < Γ i ( t ) Γ i ( t + t ′ ) > = Q i δ i j δ ( t ′ ) <\Gamma_i(t)>=0,<\Gamma_i(t)\Gamma_i(t+t')>=Q_i\delta_{ij}\delta(t') <Γi(t)>=0,<Γi(t)Γi(t+t′)>=Qiδijδ(t′)
< Γ i ( t ) η j ( t ) > = 0 <\Gamma_i(t)\eta_j(t)>=0 <Γi(t)ηj(t)>=0
F i F_i Fi和 Φ i j \Phi_{ij} Φij可以为线性或者非线性。需要设计算法，从数据中重构出网络连接。假设在整个网络中我们只能够测量两个节点 A A A和 B B B，并且有充足的数据。
x A ( t ) = [ x A ( t 1 ) , x A ( t 2 ) . . . . , x A ( t k ) , . . . , x A ( t L ) ] x_A(t)=[x_A(t_1),x_A(t_2)....,x_A(t_k),...,x_A(t_L)] xA(t)=[xA(t1),xA(t2)....,xA(tk),...,xA(tL)]
x B ( t ) = [ x B t 1 ) , x B ( t 2 ) . . . . , x B ( t k ) , . . . , x B ( t L ) ] x_B(t)=[x_Bt_1),x_B(t_2)....,x_B(t_k),...,x_B(t_L)] xB(t)=[xBt1),xB(t2)....,xB(tk),...,xB(tL)]
0 < Δ t = t k + 1 − t k ≪ 1 , L ≫ 1 0<\Delta t=t_{k+1}-t_k\ll 1,L\gg 1 0<Δt=tk+1−tk≪1,L≫1
对 x i ( t ) x_i(t) xi(t)求时间的二阶段导可得
x ¨ i ( t ) = ∂ F i [ x i ( t ) ] ∂ x i ( t ) x ˙ i ( t ) + ∑ j = 1 , j ≠ i N ∂ Φ i j [ x i ( t ) , x j ( t − τ i j ) ] ∂ x i ( t ) x ˙ i ( t ) + ∑ j = 1 , j ≠ i N ∂ Φ i j [ x i ( t ) , x j ( t − τ i j ) ] ∂ x j ( t − τ i j ) x ˙ j ( t − τ i j ) + η ˙ i ( t ) + Γ ˙ i ( t ) \ddot{x}_i(t)=\frac{\partial F_i[x_i(t)]}{\partial x_i(t)}\dot{x}_i(t)+\sum_{j=1,j\neq i}^N\frac{\partial\Phi_{ij}[x_i(t),x_j(t-\tau_{ij})]}{\partial x_i(t)}\dot{x}_i(t)+\sum_{j=1,j\neq i}^N\frac{\partial\Phi_{ij}[x_i(t),x_j(t-\tau_{ij})]}{\partial x_j(t-\tau_{ij})}\dot{x}_j(t-\tau_{ij})+\dot{\eta}_i(t)+\dot{\Gamma}_i(t) x¨i(t)=∂xi(t)∂Fi[xi(t)]x
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