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###Controlling statistical moments of stochastic dynamical networks Dmytro Bielievtsov, Josef Ladenbauer, and Klaus Obermayer DOI: 10.1103/PhysRevE.94.012306

The aim of the work is to explore how the first and second statistical moments of stochastic dynamical networks can be controlled by interfering only with a subset of nodes. The approach was demonstrated on a stochastic Hopfield network and a stochastic whole brain network model of 66 nodes obtained human diffusion imaging and functional magnetic resonance imaging data at rest.The simulations were performed using Python with the libraries “SciPy” and “Theano”.

The described approach, known as pinning control (or clamping control), consists of the following steps:

Description of first and second statistical moments of a system; 2.
- Identification of moments’ subsets which can steer the whole system in a desired state;
- Identification of original nodes corresponding to described moments’ subsets;
Feedback controller design.

Step 1

The network is described by the following equation: $\dot{x}=f(t,x) + M\eta(t)$ f(t,x) is a direct graph of the system with N nodes, and M is a constant noise mixing matrix. The moment system (MS) described by the following equations:

$\dot{\mu}=f(t,\mu) +\frac{1}{2}\nabla\nabla f(t,\mu)C$

$\dot{C}=\nabla f(t,\mu)C + C \nabla f(t,\mu) + Q$

where $\nabla f(t,\mu)$ is the Jacobian of f and Q is the total noise covariance matrix.

Step 2

A subset K of the vertex set I := {1,…,N} for the system is a set of switching nodes(SN), if the network is guaranteed to converge to a solution with SN are forced to attain the values of one of the solution when t tends to infinity. For that the following requirements should be sutisfied:

Assumption 1 For all initial conditions state of the system stays within a ball of finite radius as time tends to infinity.
Assumption 2 the Jacobian matrix of f(t, x_i, x_I) is strictly negative definite. If assumtions fullfilled that the set of SNs is identical to the feedback vertex set (FVS) of the graph of the system.

The set of SNs ( $K^*$ ) consists of the means and variances of the nodes in K and the covariances between these nodes and all others:

$K^*=\{\mu_i, C_{i,j}, C_{j,i}: j\in K, i\in I\}$

Step 3

The goal of this step is to design a stochastic feedback controller wich drives the system to a target metastable space in the moments space.

Linear feedback controller is described as:

$u_K (t, x_J (t) ):=\[\mu_g (t) C^{\frac{1}{2}}_g (t)\nu_g (t)\] + W(t)^T x_J (t)$

where $[\mu_g (t) C^{\frac{1}{2}}_g (t)\nu_g (t)\] )$ is an uncorrelated Gaussian process with mean and covariance $\mu_g (t), C_g (t)$ . W(t) ia a feedback weighting matrix to be determined with:

$W(t)=C^{-1}_{JJ}C^*_{JK}(t) )$

Taking pinning into account yields:

$\dot{\mu_J}=f_J + \frac{1}{2}\nabla\nabla f_JC$ ,

$\dot{C_JJ}=\nabla_J f_JC_JJ + C_JJ\nabla_J f^T_J + Q_JJ + \nabla_K f_JC_KJ + C_KJ \nabla_K F^T_J)$

NOTE! The descibed control system depends on a state vector $x_J(t)$ evaluated at $x-\mu$ . This implies full observability of the system!