# Complex Network Analytics for Structural-Functional Decoding of Neural Networks
## 4 Experiment and Result Analysis
### Experimental Setup
#### 4.1.2 Evaluation Index  
Compared with traditional neural network performance evaluation methods (such as accuracy, loss function value, etc. ), the analysis method based on complex network feature volume has unique theoretical advantages: firstly, by mapping the neural network parameter space into a graph structure, it can transform the black box operation, which is difficult to be explained in the field of deep learning, into a topological feature of clear physical significance in the field of network sciencecomparability; secondly, the complex network metrics are comparable across models, which can break through the inherent differences of different network architectures and reveal the intrinsic role of the regularisation mechanism; furthermore, the method supports fine-grained tracking of the training dynamics, which can capture the phenomenon of critical phase transition emerging during the weight evolution. This structure-driven analysis method provides a new decoupling perspective for understanding the core issues such as generalisation ability and robustness of neural networks. In order to quantitatively analyse the deeper characteristics of the model structure, this study constructs a multi-dimensional complex network feature index system, covering six core indexes in two categories and incorporating basic topological features, statistical features of weights, and special operation analysis:  
(1) Overall Topological Characteristics  
The underlying topology characteristics metrics focus on the basic attributes of the network connectivity patterns, including node/edge size, node degree, network density, and average clustering coefficient. 1. Number of nodes/edges: The number of nodes and edges is the number of nodes and connected edges of a complex network counted after the neural network is transformed into a complex network. The number of nodes indicates the absolute number of neurons in the network, reflecting the model complexity, and the number of connected edges indicates the total number of trainable parameters of the whole network, characterising the model capacity. 2. Network density: The ratio of the actual number of connections to the theoretical maximum number of connections, reflecting connection sparsity and calculated as the ratio of the actual number of edges to the maximum possible number of edges, is as follows:  
\[D=\frac{E}{N(N-1)} \tag{12}\]  
3. Average clustering coefficient: Used to measure the closeness of connections between node neighbours and to assess the modularity properties of the network, it is calculated as the ratio of the actual number of triangular connections in a node's neighbourhood to the theoretical number of possibilities, as shown in Equation (13):  
\[C=\frac{1}{N}\sum_{i=1}^{N}\frac{T_{i}}{k_{i}(k_{i}-1)} \tag{13}\]  
where \(T_{i}\) is the actual number of connections between neighbours of the node \(v_{i}\) and \(k_{i}\) is the node degree. (2) Individual node characteristics  
1. Node degree: captures the distribution of individual node weights. Average number of connections per node is used as a measure of network connectivity activity. 2. Input/Output Strength Difference This calculates the ratio of the sum of node input weights to the sum of node output weights. If the Gini coefficient of node strength is closer to 1, this indicates that a few hub nodes are carrying most of the information flow and the network tends to be scale-free structure. 3. Sparse connection ratio This is the percentage of weights whose absolute value is less than a certain threshold. In addition, this study introduces the hierarchical index to comprehensively assess the multi-scale abstraction capability of the network, which quantifies the degree of structuring of the network from local feature perception to global semantic integration by analysing the results of hierarchical clustering of node strengths.