# Percolation Distribution in Small-World Quantum Networks
## 3 Percolation Optimization  
In order to ensure that any node in the network can be connected through entanglement swapping after percolation, we add a central node in the center of the network, which is connected to other nodes in the network. Taking a hexagonal network in a cellular network as an example, as shown in Figure 4. In this case, fewer isolated nodes are made after entanglement percolation and most of the nodes in the network can communicate with others through entanglement swapping. Partial q-swap (q-pswap) operation is shown in Figure 5. For a node with a degree of four, entanglement swapping is only performed on part of the connected edges, and the rest of the connected edges can be swapped with the neighboring edges of other nodes, which can ensure that some important nodes do not become isolated nodes. At the same time, the impact of q-swap of one node on other nodes can be reduced. The variation in GCC with edge conversion probability \(p\) is an important property in the process of entanglement percolation of the network. For small-world quantum networks, there are several important statistical properties [5]: clustering coefficient, degree distribution, and average shortest path. The clustering coefficient is a measure of the tendency of the nodes in a network to cluster together. The shorter the average shortest path, the fewer the number of entanglement swapping and resources are required by the nodes in the network to establish communication links. We choose a WS quantum network with the number of nodes \(N=2000\), the reconnection probability of the network \(p_{r}=0.4\), and the average network degree \(\bar{k}=4\). After adding the central node, the average degree of the network becomes 6 (\(\bar{k}=6\)). In the  
Figure 4: **(a)** Add the central node (red circle) to the network of 6 nodes (blue circle), the central node has connected edges with any other node in the network. **(b)** Perform a disconnect–reconnect operation on the network, no operation is performed on the connected edges of the central node with other nodes during the process. Figure 3: Effects of 4-swap on the structure of the underlying networks. process of disconnection-reconnection and quantum preprocessing, the connected edges of the central node are not involved in any operation. The central node is only used as an auxiliary node to ensure that there are not too many isolated nodes in the network and to reduce the average shortest path of the network and the resources required to establish connections. In the preprocessing process, we perform 4-swap as well as 4-pswap operations on nodes with a degree of 4. The degree distribution is shown in Figure 6. Due to the existence of the central node, no isolated nodes will appear in the network. However, in the process of percolation, the connected edges of the central node will participate in the process, resulting in isolated nodes in the network. However, compared to the classical WS quantum networks, the addition of the central node can still reduce the appearance of isolated nodes. After 4-pswap, the degree distribution of the network is more uniform, and the probability of nodes with large degree values appears higher. In the process of percolation, nodes with larger degree values are more likely to percolate successfully. Figure 5: Different quantum preprocessing methods. (**a**) The central node (red circle) with a degree of 4 can perform entanglement swapping on 4 connected edges when doing quantum preprocessing. (**b**) The network structure obtained by doing q-swap on the central node, and the peripheral nodes (black circles) are not involved in the quantum preprocessing. (**c**) The central node operates on only 3 of its connected edges, and the last one is entangled and swapped with the connected edges of the peripheral nodes to obtain a new connected edge (red line). All nodes in the network are involved in the preprocessing process. Figure 6: The degree distribution of the network before and after quantum preprocessing.