【学术】重构具有时间延迟相互作用的动力学网络

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

理论推导

考虑$N$个节点的动力学系统
$\dot{x}_i=F_i[x_i(t)]+\sum_{j=1,j\neq i}^N\Phi_{ij}[x_i(t),x_j(t-\tau_{ij})]+\eta_i(t)+\Gamma_i(t),i=1,2,...,N$

其中$\eta_i(t)$为色噪声，$\Gamma_i(t)$为白噪声，满足
$<\eta_i(t)>=0,<\eta_i(t)\eta_i(t+t')>=P_{ij}e^{-\frac{|t'|}{\tau_C}}$

$<\Gamma_i(t)>=0,<\Gamma_i(t)\Gamma_i(t+t')>=Q_i\delta_{ij}\delta(t')$

$<\Gamma_i(t)\eta_j(t)>=0$

$F_i$和$\Phi_{ij}$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$A$和$B$，并且有充足的数据。
$x_A(t)=[x_A(t_1),x_A(t_2)....,x_A(t_k),...,x_A(t_L)]$

$x_B(t)=[x_Bt_1),x_B(t_2)....,x_B(t_k),...,x_B(t_L)]$

$0<\Delta t=t_{k+1}-t_k\ll 1,L\gg 1$

对$x_i(t)$求时间的二阶段导可得
$\ddot{x}_i(t)=\frac{\partial F_i[x_i(t)]}{\partial x_i(t)}\dot{x}_i(t)+\sum_{j=1,j\neq i}^N\frac{\partial\Phi_{ij}[x_i(t),x_j(t-\tau_{ij})]}{\partial x_i(t)}\dot{x}_i(t)+\sum_{j=1,j\neq i}^N\frac{\partial\Phi_{ij}[x_i(t),x_j(t-\tau_{ij})]}{\partial x_j(t-\tau_{ij})}\dot{x}_j(t-\tau_{ij})+\dot{\eta}_i(t)+\dot{\Gamma}_i(t)$

其中高阶导数可以使用前向差分进行计算
$\dot{x}_i(t_k)=\frac{x_i(t_{k+1})-x_i(t_{k})}{\Delta t},\ddot{x}_i(t_k)=\frac{\dot{x}_i(t_{k+1})-\dot{x}_i(t_{k})}{\Delta t}$

噪声的导数被定义为
$\dot{\eta}_i(t_k)=\frac{\eta_i(t_{k+1})-\eta_i(t_{k})}{\Delta t},\dot{\Gamma}_i(t_k)=\frac{\Gamma_i(t_{k+1})-\Gamma_i(t_{k})}{\Delta t}$

根据前面的公式可知
$\ddot{x}_A(t_k)=\frac{1}{2}\frac{\partial F_A[x_A(t_k)]}{\partial x_i(t_k)}[\dot{x}_A(t_k) +\dot{x}_A(t_{k+1})]+\frac{1}{2}\sum_{j=1,j\neq A}^N\frac{\partial\Phi_{Aj}[x_A(t),x_j(t_k-\tau_{Aj})]}{\partial x_A(t_k)}[\dot{x}_A(t_k)+\dot{x}_A(t_{k+1})] +\frac{1}{2}\sum_{j=1,j\neq A}^N\frac{\partial\Phi_{Aj}[x_i(t),x_j(t-\tau_{Aj})]}{\partial x_j(t-\tau_{Aj})}[\dot{x}_j(t_k-\tau_{Aj})+\dot{x}_j(t_{k+1})] +\frac{\eta_A(t_{k+1-\tau_{Aj}})-\eta_A(t_{k})}{\Delta t} +\frac{\Gamma_A(t_{k+1})-\Gamma_A(t_{k})}{\Delta t}$

然后对方程左右两边同乘$x_B(t_k+\Delta t)$，计算每项的关联可得
$R_{AB}=<\ddot{x}_A(t_k)x_B(t_k+\Delta t)>$

$=<\frac{1}{2}\frac{\partial F_A[x_A(t_k)]}{\partial x_i(t_k)}[\dot{x}_A(t_k)x_B(t_k+\Delta t)+\dot{x}_A(t_{k+1}x_B(t_k+\Delta t))]>$

$+\frac{1}{2}\sum_{j=1,j\neq A}^N\frac{\partial\Phi_{Aj}[x_A(t),x_j(t_k-\tau_{Aj})]}{\partial x_A(t_k)}[\dot{x}_A(t_k)x_B(t_k+\Delta t)+\dot{x}_A(t_{k+1})x_B(t_k+\Delta t)]$

$+\frac{1}{2}\sum_{j=1,j\neq A}^N\frac{\partial\Phi_{Aj}[x_i(t),x_j(t-\tau_{Aj})]}{\partial x_j(t-\tau_{Aj})}[\dot{x}_j(t_k-\tau_{Aj})x_B(t_k+\Delta t))+\dot{x}_j(t_{k+1})x_B(t_k+\Delta t))]$

$+\frac{\eta_A(t_{k+1-\tau_{Aj}})x_B(t_k+\Delta t)-\eta_A(t_{k})x_B(t_k+\Delta t)}{\Delta t}+\frac{\Gamma_A(t_{k+1})x_B(t_k+\Delta t)-\Gamma_A(t_{k})x_B(t_k+\Delta t)}{\Delta t}$

记该公式为方程1。因为有白噪声的存在，对$<\dot{x}_i(t)\dot{x}_j(t+t')>$积分在$t'=0$处有个阶跃，所以
$<\dot{x}_i(t)x_j(t+\Delta t)>-<\dot{x}_i(t)x_j(t)>=Q_j\delta_{ij}$

定义$n_{AB}=\frac{\tau_{AB}}{\Delta t}$，$V_i(\Delta k)=<\dot{x}_i(t_k)x_i(t_{k+\Delta k})>$，在$\Delta k=0$到$\Delta k=1$处有
$V_i(1)-V_i(0)=Q_i$

由于$<\dot{x}_B(t_k-\tau_{AB})x_B(t_{k+\Delta k})>$和$<\dot{x}_B(t_{k+1}-\tau_{AB})x_B(t_{k+\Delta k})>$在$k=-n_{AB}$处不联系，可以得到$n_{AB}$即$\tau_{AB}$
根据$<\dot{x}_i(t)x_j(t+\Delta t)>-<\dot{x}_i(t)x_j(t)>=Q_j\delta_{ij}$的性质可知方程1右边除了$<\dot{x}_B(t_k-\tau_{AB})x_B(t_{k+\Delta k})>$和$<\dot{x}_B(t_{k+1}-\tau_{AB})x_B(t_{k+\Delta k})>$没有不连续的项，所以从方程1可以得到
$D_{AB}=R_{AB}(-n_{AB}+2)-R_{AB}(-n_{AB})=<\ddot{x}_A(t_k)x_B(t_{t_k-n_{AB}+2})>-<\ddot{x}_A(t_k)x_B(t_{t_k-n_{AB}})>$

$=<\frac{\partial\Phi_{Aj}[x_i(t),x_j(t-\tau_{Aj})]}{\partial x_B(t-\tau_{AB})}>Q_B$

定义$M_{AB}=<\frac{\partial\Phi_{Aj}[x_i(t),x_j(t-\tau_{Aj})]}{\partial x_B(t-\tau_{AB})}>$，可得
$M_{AB}=\frac{D_{AB}}{Q_B}=J_{AB}$

数值仿真

定义均方根误差$E$作为衡量$M_{AB}$和$J_{AB}$之间的误差，$T$表示测量时长
对于线性系统：

对于扩散耦合的FHN网络（上），和Rossler网络（下）

对于基因调控网络

总结

文章的核心亮点是重构有时滞作用的系统，trick是利用白噪声的性质。