# Complex Network Analytics for Structural-Functional Decoding of Neural Networks
## 3 Structural-Functional Decoding Model
### Regularising Virtual Nodes  
(1) BatchNorm  
The BatchNorm layer does not generate nodes directly; its functionality is achieved by adjusting the weight attributes of the predecessor edges. This design is based on the nature of the operation of the normalisation layer, a linear transformation of the output of the precursor layer, rather than the introduction of new computational units [29]. Specifically, the scaling factor \(\gamma\) and the offset \(\beta\) of each BatchNorm layer are fused into the computation of the weights of the corresponding input edges. The original value of the edge weights from the convolutional layer to the BatchNorm layer is \(w_{ij}\). Adjustment of the edge weights is achieved by affine transformation, which can be formalised as follows:  
\[w_{ij}^{{}^{\prime\prime}}=\gamma\cdot w_{ij}^{{}^{\prime}}+\beta \tag{5}\]  
where \(w_{ij}^{{}^{\prime}}\) is the normalised value of the edge weights, and this manipulation encodes the feature distribution normalisation process into the edge weight dynamics, allowing the complex network to reflect the moderating effect of normalisation on signal transmission.  
From the point of view of complex network dynamics, the BatchNorm treatment accurately reproduces the statistical normalisation process in the normalisation layer at the topological level, and the evolution of the weights of the relevant edges can show the model training convergence stability. This effect makes the distribution of edge weights more Gaussian and reduces the probability of the occurrence of extreme weights, which is manifested in the reduction in the variance of the edge weights of the complex network.  
(2) Dropout  
The Dropout layer also does not create new nodes, and its role is realised by adding probabilistic mask attributes to subsequent connected edges. This design corresponds precisely to the operational semantics of Dropout, randomly disconnecting neurons rather than modifying the computational unit itself [30]. Let the original edge weights be \(w_{ij}\), the Dropout retention probability be \(p\), and the disconnection probability be \(1-p\). The Dropout layer behaves as an overall compression of the connecting edge weights in a complex network, which can be formalised as follows:  
\[w_{ij}^{{}^{\prime}}=\frac{m_{ij}w_{ij}}{p} \tag{6}\]  
where \(m_{ij}\) is a mask variable obeying a Bernoulli distribution. This operation is equivalent to continuous subgraph sampling of the network, where continuously surviving edges tend to correspond to critical feature transfer paths, while frequently disconnected edges are redundant connections.  
From the perspective of complex network analysis, the random disconnections of Dropout affect the fluctuation of the weights of the connected edges, and their contribution to the robustness of the model can be analysed by quantifying the variation in the variance of the weights of different connected edges. This structure-dynamics correlation provides a theoretical basis for analysing the impact of Dropout on the structure of complex networks, and by calculating the covariance between the survival probability of edges and the gain in weights; connectivity paths that are critical to the generalisation ability of the model can be identified.