## 4 Results
### Theory Analysis  
According to the related content of reference [32], using related knowledge of continuous medium theory, mean field theory, and algebraic theory, the distribution law of node strength in the constructed network model is analyzed, and the validity of the model is analyzed from a theoretical level. Assumptions: The node strength \(s_{i}\) is continuously changed, and the evolution time \(t\) is also continuously changed. According to the implementation flow, the change rate of \(s_{i}\) is deduced as follows. According to the step in (a) in Section 3.2.3, the edge-weight caused by newly added nodes evolves to:  
\[ \begin{array}{ll}\frac{dw_{ij}(t)}{dt}&=mp_{1}\prod\limits_{local(v_{i})} \alpha\frac{w_{ij}}{s_{i}(t)}+mp_{1}\prod\limits_{local(v_{j})}\alpha\frac{w_ {ij}}{s_{j}(t)}\\ &=mp_{1}\frac{M}{N(t)}\Big{(}\frac{s_{i}(t)}{\sum_{k\in local}s_{k}(t)}\alpha \frac{w_{ij}(t)}{s_{i}(t)}+\frac{s_{j}(t)}{\sum_{k\in local}s_{k}(t)}\alpha \frac{w_{ij}(t)}{s_{j}(t)}\Big{)}\\ &=2 mp_{1}\alpha\frac{M}{N(t)}\frac{w_{ij}(t)}{\sum_{k\in local}s_{k}(t)}\end{array}  \tag{16}\]  
and the rate of change of \(s_{i}\) is  
\[ \begin{array}{ll}\left(\frac{ds_{i}}{dt}\right)_{1}&=mp_{1}\frac{M}{N(t)} \frac{s_{i}}{\sum_{k\in local}s_{k}(t)}+\frac{dw_{ij}(t)}{dt}\\ &=mp_{1}(1+2\alpha)\frac{M}{N(t)}\frac{s_{i}(t)}{\sum_{k\in local}s_{k}(t)} \end{array}  \tag{17}\]  
According to the step in (b) in Section 3.2.3, and because the connection is bidirectional, the corresponding edge weights will evolve according to the following equation:  
\[\frac{dw_{ij}(t)}{dt}=\frac{s_{j}(t)}{\sum_{k\in local}s_{k}(t)-s_{i}(t)}\frac {s_{i}(t)}{\sum_{k\in local}s_{k}(t)-s_{j}(t)} \tag{18}\]  
and the rate of change of \(s_{i}\) is  
\[ \begin{array}{ll}\left(\frac{ds_{i}}{dt}\right)_{2}&=m_{1}p_{2}\frac{M}{N(t )}\left[\frac{1}{M}+\left(1-\frac{1}{M}\right)\sum\limits_{j\in local}\frac{ dw_{ij}(t)}{dt}\right]\\ &\approx m_{1}p_{2}\frac{M}{N(t)}\left[\frac{1}{M}+\left(1-\frac{1}{M}\right) \frac{s_{i}(t)}{\sum_{k\in local}s_{k}(t)-s_{i}(t)}\right]\\ &\approx m_{1}p_{2}\frac{M}{N(t)}\left[\frac{1}{M}+\left(1-\frac{1}{M}\right) \frac{s_{i}(t)}{\sum_{k\in local}s_{k}(t)}\right]\end{array}  \tag{19}\]  
According to the step in (c) in Section 3.2.3, the rate of change of \(s_{i}\) is:  
\[\left(\frac{ds_{i}}{dt}\right)_{3}=m_{2}p_{3}\frac{M}{N(t)}\frac{s_{i}(t)}{ \sum_{k\in local}s_{k}(t)}+m_{2}p_{3}\bigg{(}1-\frac{M}{N(t)}\bigg{)}\frac{s_{ i}(t)}{\sum_{k\
otin local}s_{k}(t)} \tag{20}\]  
For [PERSON]'s expansion of the exponential function in Equation (29), because s can be seen as an infinite function of \(t\), \(s\gg t\), \(s\gg m\).When \(t\) is large enough, the intensity distribution of node \(v_{i}\) is similar to:  
\[p(s)\approx\frac{\beta t}{a}\frac{\left(m+\frac{b}{a}\right)^{\frac{1}{2}}}{ \left(s+\frac{b}{a}\right)^{\frac{1}{2}+1}} \tag{30}\]  
among them, the power law distribution index is  
\[\gamma_{s}=\frac{1}{a}+1=\frac{2[mp_{1}(1+\alpha)+m_{1}p_{2}+m_{2}p_{3}-m_{3}p _{4}]\beta}{mp_{1}(1+2\alpha)+m_{1}p_{2}\left(1-\frac{1}{M}\right)+2m_{2}p_{3}+ m_{3}p_{4}\left(1-\frac{1}{M}\right)}+1 \tag{31}\]  
Analysis: In Equation (31), assign different values to each probability, and the corresponding models are different:  
1. When \(p_{1}=1,\ p_{2}=0,\ p_{3}=0,\ p_{4}=0,\ \beta=1\), and at this time, \(\gamma_{s}=\frac{1}{a}+1=\ \frac{4a+3}{2\alpha+1}\). At this point, \(s_{i}(t)\) is the change rate of the node strength of the BBV model. 2. When \(p_{1}=1,\ p_{2}=0,\ p_{3}=0,\ p_{4}=0,\ \beta=1,\ \alpha=0\), edge weights no longer change, and this network is non-weighted networks. At this time, \(s_{i}=k_{i}\) for all nodes, and \(s_{i}(t)\) is the change rate of the node degree of the BA model. 3. When the time \(t\) is large enough, the node intensity distribution of the model is a power law distribution, and the distribution index is \(\frac{1}{a}+1\).