# Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation
## 2 Results
### Improving the Overall Quality of a Complex Cluster  
We test here cluster states with the structure of the three models defined above: Barabasi-Albert (BA), Erdos-Renyi (ER), and Watts-Strogatz (WS), with different characterizing parameters \(m_{BA}\), \(p_{ER}\), and \(p_{WS}\). In Figure 2, the difference between a scale-free network and a random network with a comparable average degree is shown. As we see in Figure 2a, the Barabasi-Albert network exhibits highly connected nodes (_lubs_) that are absent in the Erdos-Renyi model (Figure 2b). We consider at first the implementation of 48-mode complex clusters. As explained above, the clusters are implemented from a 48-mode squeezed vacuum via the application of the linear optics unitary of Equation (3) corresponding to the graphical shape we want to reach [25]. The scheme  
Figure 1: Rewiring of a regular 48-node network for the construction of a “small world” network as shown in [35]. Figure 2: Comparison between two models of complex networks, both with an average degree of \(\langle k\rangle\sim 3.9\). The size of the dots increases with the number of links. (**a**) Barabási–[PERSON] model with \(m_{BA}=2\), with a maximum node degree of \(k=22\); (**b**) Erdős–[PERSON] model with \(p_{ER}=4/49\), with a maximum node degree of \(k=8\). is presented in Figure 3a. In order to pursue realistic implementations, we consider the list of squeezing values presented in Figure 3b, which corresponds to a series of faithful values that can be obtained via the Schmidt decomposition of parametric process Hamiltonian of the experiment described in [20; 37]. In this work, the input squeezing values for the implementation of the cluster are fixed: this is because our interest lies in understanding what is the best we can do when, like in a realistic situation, we are provided with a given set of squeezed input states. We stress that this scheme, as well as the associated results, is different to the one dealing with the canonical decomposition of [22]. The obtained clusters are optimized, acting on the parameters of the arbitrary orthogonal matrix \(\mathcal{O}\) of Equation (3), by minimizing the fitness function \(f(\Delta^{2}\delta_{i})=\sum_{i}\Delta^{2}\delta_{i}\), where \(\Delta^{2}\delta_{i}\) denotes the \(i\)-th nullifier normalized with the vacuum noise. As BA, ER, and WS are statistical models for graph families, we implement \(N=100\) graphs for each model with a given set of parameters. We define the average quality of a single graph \(j\) as \(\mu_{j}\) and the average quality for a set of graphs as \(\mu\), as follows:  
\[\mu_{j}=\frac{1}{48}\sum_{i}\Delta^{2}\delta_{i,j}\ ;\quad\mu=\langle\mu_{j}\rangle, \tag{4}\]  
where \(\Delta^{2}\delta_{i,j}\) identifies variance of the \(i\)-th nullifier of the \(j\)-th graph. We will indicate as \(\sigma\) the standard deviation of the \(\mu_{j}\) values. This way, we can average out the fluctuations due to the randomness of the complex shape. As we can see from Table 1, the implementation of quantum complex networks following the Barabasi-Albert model or the Erdos-Renyi model shows that the quality of the cluster increases when the average number of edges per node, the average degree \(\langle k\rangle\), increases. The average degree \(\langle k\rangle\) can be raised by increasing the parameters \(m_{BA}\) and \(p_{ER}\), for the BA and the ER models, respectively. The results clearly show that the quality of the cluster \(\mu\) increases with these parameters, until the limiting case of the fully connected graph (\(m_{BA}=47\) for the Barabasi-Albert model and \(p_{ER}=1\) for the Erdos-Renyi model) is reached. The Watts-Strogatz cluster confirms that the quality of the cluster increases for a larger value of \(\langle k\rangle\), as we can see by comparing Table 2a,b, but it also shows a peculiar behavior that depends on \(p_{WS}\).