# Complex Network Analytics for Structural-Functional Decoding of Neural Networks
## 3 Structural-Functional Decoding Model
### Convolutional Layer Formal Network Modelling  
The working process of the convolutional layer is described as the \(k\)-th output channel of the convolutional kernel, whcih is a spatial convolution operation performed on the input feature map \(\chi\) by means of an independent 3D tensor \(W^{k}\in\mathbb{R}^{c_{in}\times h\times w}\)[22], as shown in Equation (1).  
\[F^{k}=\sum_{c=1}^{c_{in}}W^{k}_{c}*\chi_{c}+b^{k} \tag{1}\]  
where \(*\) denotes the convolution operation and \(b^{k}\) is the channel bias term.  
(1) Node description  
Considering the function separation characteristic of convolution operation, the network modelling of a convolution layer adopts the principle of channel-level node mapping [1]. Each output channel of the convolution kernel is essentially an independent feature detector, and the response mode of the complex network node \(v_{k}\) inherits the feature detection function of the original channel; each output channel \(F^{k}\) is mapped as an independent node \(v_{k}\) in the complex network, and the total number of nodes is equal to the number of output channels \(c_{out}\), \(V=\{v_{k}|1\leq k\leq c_{out}\}\).  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**NN Component** & **Graph Node** & **Graph Edge** \\ \hline Convolutional Layer & Output channels & Input\(\rightarrow\)output channel \\ Fully Connected Layer & Neurons & All input\(\rightarrow\)output connections \\ Pooling Layer & Aggregation units & Local receptive fields \\ BatchNorm/Dropout & Virtual nodes (no mapping) & Edge weight modulation \\ \hline \hline \end{tabular}
\end{table}
Table 1: The mapping from neural network components to graph structures.  
Figure 1: The diagram of the transformation between neural networks and complex networks.  
2. Edge description The edge connection pattern of the convolutional layer is determined by the 3D convolutional kernel tensor, which establishes a single edge connection between the input and output channels, and only a single edge is generated for each input channel [23]; on each input channel \(\chi_{c}\), the convolutional kernel window \(W_{c}^{k}\) creates \(h\times t\) edges for \(v_{k}\), \(E=\left\{e_{ij}|i\in[1,c_{in}],j\in[1,c_{out}]\right\}\) when the spatial dimensions are sliding windowed, where \(e_{ij}\) denotes the directed edges from the input channel \(i\) to the output node \(v_{j}\). (3) Edge weight design Each input channel node and the corresponding output channel node are connected by the spatial weights in the local sensory field; the node activity in the complex network can directly correspond to the channel activation strength of the feature map in the neural network [3] and the weights of the edges are calculated by using the absolute value of the mean value of the convolution kernel slices, as shown in Equation (2). \[w_{ij}=\frac{1}{ht}\sum_{x\leq h}\sum_{y\leq t}\left|K_{ij}(x,y)\right|\] (2) where \(K_{ij}\) denotes the convolution kernel connecting the input channel \(i\) to the output channel \(j\). This calculation is equivalent to extracting the energy spectral density of the convolution kernel, highlighting the spatial pattern quantisation that contributes significantly to the feature transfer, reducing the computational complexity through spatial averaging and eliminating the interference of the parameter polarity on the assessment of the connection strengths through the absolute value processing.