## 5 Experimental Section
### Experiments about L1 Optimization with MATLAB
#### 5.1.2 The Effect of L1 Optimization in Network Random Coding  
We will investigate the performance of the Algorithm 1. The ratio of the messages sent through the secret channel is denoted as \(\vartheta\). \(\|x_{0}\|\) is on behalf of the sparseness of vector \(x_{0}\). In Algorithm 1, if the components of the original uncompressible messages \(\kappa\) have no zeroes, \(\|x_{0}\|=k/n\). In Figure 6, we set \(\|x_{0}\|=1\).  
Most of the parameters are the same as the simulation in [21]. The parameters are as follows. \(v=0.05\), \(\delta=0.25\). \(m\in\{100,\,200,\,400,\,800\}\). We have not got any channel to get the original implement details of [21]. Because some implement details may be different, our effectiveness is a little worse. However, the whole trend is the same.  
In Figure 9, \(\|x_{0}\|=1\) and \(\vartheta\) is set as 0, 0.1, 0.2 and 0.3 respectively in different subfigures. In Figure 8(a), we can see that the percentage of successful recovery and the fraction of corrupted errors are in inverse proportion. When \(m\) increases, the correcting fraction \(\tau\) also increase, and almost approaches 1. This point is surprising and attractive. It can be natural to adopt this L1 optimization for correcting the dense propagated errors in network coding. In Figure 8(a), even 0.95 density errors can be recovered. However, the successful correction fraction is not satisfactory when errors density is high, for example, 0.95. However, we can increase \(m\) to increase the successful correction ratio. Generally, when \(m=800\) and fraction of errors is 0.6, the fraction of successful correction approaches 1. This certainly can meet the need in real communication. However, \(\|x_{0}\|=1\) also means a low information rate here. However, when the fraction of corrupted errors is 100%, this algorithm cannot recover the original uncompressible messages. In Figure 8(b)-d, the secret channel is used. With the increase of \(\vartheta\), we can correct more dense errors. Especially, when the fraction of propagated errors is 100%, we can also correct it.  
If \(\|x_{0}\|=1\), the information rate will be very low. We will also investigate the performance of the Algorithm 1 at different \(\|x_{0}\|\). In Figure 10, \(\|x_{0}\|=m^{1/2}\). The high fraction of successful decoding is at the cost of the low information rate. If we want to increase the information rate, the fraction of successful decoding will be down. However, even when the information rate is higher, L1 optimization also has a surprisingly high fraction of corrected errors. In traditional codes, the fraction of corrected errors is 0.5 at most when the information rate approaches 0. The fraction of successful decoding is approximately 0.47. It is also higher than traditional codes, i.e., to a considerable information rate.  
Figure 8: The spread of errors in non-coherent networks on field size. The blue line is the function of spread normalized error rate based on the size of field.  
Figure 10: Error correction in L1 optimization when \(\|x_{0}\|=m^{1/2}\) at different \(\theta\). (a) \(\theta=0\); (b) \(\theta=0.1\); (c) \(\theta=0.2\); (d) \(\theta=0.3\).  
Both \(\|x_{0}\|=1\) and \(\|x_{0}\|=m^{1/2}\) are extreme situations. It would be better to keep a balance between error-correcting ratio and information rate. It shows, when \(\|x_{0}\|\) increases, i.e., higher information rate, the fraction of correction becomes lower. However, the fraction of correction is also acceptable.