# Complex Network Analytics for Structural-Functional Decoding of Neural Networks
## 3 Structural-Functional Decoding Model
### Fully Connected Layer  
The complex network modelling process in the fully connected layer follows strict mathematical correspondence rules [24]. From the feature space transformation perspective, individual nodes essentially construct a nonlinear projection from the n-dimensional input space \(\mathbb{R}^{n}\) to the output space \(\mathbb{R}^{m}\). The form of the operation is shown in Equation (3). It can be decomposed into two phases: firstly, an affine transformation \(w^{T}x+b\) is performed on the input feature vector, and subsequently a nonlinear decision boundary is introduced through the activation function.  
\[y_{j}=\sigma\Big{(}w_{j}^{T}x+b_{j}\Big{)} \tag{3}\]  
where \(w\) is the weight vector, which defines the hyperplane direction in the feature space, \(b\) is the bias term, and \(\sigma\) is the activation function. This mapping relationship is expressed in complex networks as node \(v_{j}\) receives input signals from all nodes in the precursor layer and modulates the signal strength through the edge weights \(w_{ij}\) of the signal inputs from all nodes in the precursor layer. (1) Node modelling The node mapping of the fully connected layer uses neuron-level granularity, with each output neuron instantiated as a network node. The independence of node operations in the fully connected layer makes each neuron naturally possess the core function of a node in a complex network, which is to serve as the basic unit of information processing [25]. Therefore, the network mapping must retain the independent computational trajectory of each neuron intact, otherwise it will not be able to accurately characterise the combination law of cross-channel features. This design deeply reflects the mathematical nature of the fully connected layer, where each neuron is a high-dimensional nonlinear combination of input features. Each neuron \(y_{i}\) in the output layer is instantiated as a network node \(v_{i}\), with the total number of nodes equal to the output dimension \(d_{out}\).  
(2) Edge description  
The edge connection pattern of the fully connected layer is explicitly defined by the weight matrix \(W\in\mathbb{R}^{d_{out}\times d_{in}}\). The in-degree \(d_{in}(v_{j})=n\) of each node \(v_{j}\) is strictly equal to the number of nodes in the precursor layer, and the out-degree \(d_{out}\big{(}v_{j}\big{)}=m\) is the same as the number of nodes in the successor layer, which forms a bi-directional fully connected topology. In this paper, we establish a fully connected topology between the input and output neurons as follows:  
\[E=\big{\{}e_{ij}|\forall i\in[1,d_{in}],\forall j\in[1,d_{out}]\big{\}} \tag{4}\]  
This design transforms the global linear transformation of the fully connected layer into a fully bipartite graph structure, where each input neuron node is connected to all output neuron nodes [26], and the dense connection of the fully connected layer realises the full dimensional interaction of the feature space, which complements the function of the local sense field of the convolutional layer.  
(3) Edge weight design  
Complex network analysis is concerned with the strength of the information flow, so the weights of the fully connected edges are directly mapped directly to the weight matrix using absolute value normalisation, \(w_{ij}=|W_{ji}|\). Absolute value processing makes the edge weights characterise only the strength of the connection without distinguishing whether they are inhibitory or not, and highlights the strength assessment of the feature combinations by eliminating the parameter polarity interferences [27].