# Complex Network Analytics for Structural-Functional Decoding of Neural Networks
## 3 Structural-Functional Decoding Model
### Node Characterisation
#### 3.5.2 Dynamic Node Activation Analysis  
This study proposes a node importance assessment method based on the dynamic features of complex networks, with the core containing three progressive stages:  
(1) Activation trajectory tracker  
A training-adaptive activation trajectory sampling module is constructed, the core of which consists of a dynamic checkpoint decision maker and a gradient landscape analyser. The system intelligently models the intensity value \(s_{i}(t)\) of each node \(v_{i}\) in the time domain by monitoring the geometric properties of the loss surface in real time so as to obtain highly information-dense node activation data at the critical stage of training. A time slice sampling mechanism for the training process is established with the following time slice sampling function:  
\[S(t_{k},g_{k})=\left\{s_{i}^{(k)},\phi\left(\lambda_{\max}^{(m)},\zeta^{(m)} \right)\right\} \tag{7}\]  
\[t_{k}=\sum_{m=1}^{k}\Delta t_{m}.\varphi\left(\lambda_{\max}^{(m)},\zeta^{(m)}\right) \tag{8}\]  
\[g_{k}=diag\left(\
abla^{2}L(t_{k})\right) \tag{9}\]  
where \(\theta_{c}\) is the curvature threshold, \(\theta_{s}\) is the smoothness threshold, \(\phi(\lambda,\zeta)=\exp(-\alpha\lambda+\beta\zeta)\) is the gradient feature weight function, and \(\varphi(\lambda,\zeta)=\frac{2}{1+e^{\gamma(\lambda-\theta_{c})}}+\frac{ \zeta}{\theta_{s}}\) is the dynamic interval modulation factor. The parameter space is calculated by stochastic projection method, the maximum curvature eigenvalue \(\lambda_{\max}=eig_{\max}(\
abla^{2}L)\), \(\
abla^{2}L\) refers to the maximum eigenvalue of the Hessian matrix of the loss function \(L\), and the smoothness is \(\zeta=\frac{\lambda_{\max}}{\lambda_{\max}+\epsilon}\). Also, the preservation process includes loss trend sensing, which adjusts the base preservation interval \(\Delta t\) by calculating the loss decline rate \(v\) during the training phase window period, as shown in Equation (10).  
\[v=\frac{1}{T}\sum_{t=1}^{T}\left(l_{t}-l_{t+1}\right) \tag{10}\]  
where \(T\) represents the length of the time window, the number of consecutive training steps used to calculate the average value, and \(l_{t}\) is the loss function value at step \(t\).  
The time interval was adjusted to \(0.8\Delta t\) when the training was in a rapid decline period \(v>0.1\) and to \(1.5\Delta t\) when the plateau period \(v<0.05\).  
Decision tree for node classification  
Hierarchical criteria were used to filter redundant nodes:  
1. Stability criterion: the ADF unit root test was performed on the intensity sequence \(\left\{s_{1}^{(1)},\cdots,s_{i}^{(N)}\right\}\) of each node, and nodes that rejected the non-stationary assumption (\(p<0.05\)) were entered into the candidate set.  
2. Significance criterion: the contribution of node strength was calculated relative to the strength of the whole network:  
\[\rho_{i}=\frac{\bar{s}_{i}}{\frac{1}{N}\sum_{j=1}^{N}\bar{s}_{j}} \tag{11}\]  
where \(N\) is the total nodes and \(\bar{s}_{i}\) is the normalised strength of node \(v_{i}\).  
Nodes with \(\rho_{i}<\theta\) will be judged as redundant nodes with connection weights mapped to zero space.  
3. Persistence criterion: it is confirmed the node strength has not undergone a phase jump by sliding window \(t\)-test to verify its distributional stability.  
The advantages of this method over the static pruning scheme are as follows: (1) dynamic threshold adaptive training process evolution; (2) exclusion of temporarily inhibited nodes by statistical tests; (3) preservation of the integrity of the network topology to maintain gradient flow stability.