# An Investigation of Particle Swarm Optimization Topologies in Structural Damage Detection
## 3 Particle Swarm Optimization
### PSO Topologies  
The PSO topologies describe the neighbor relationship and interaction between particles, which can control the propagation of information in the particle swarm and directly affect the particle swarm's optimization ability and convergence. The PSO topologies can be divided into two categories: static topologies and dynamics topologies. For static topologies, each particle's neighborhood does not change in the whole optimization process; for dynamics topologies, the neighborhoods of some individuals vary during the iterations. Subsequently, some commonly used static topologies and dynamic topologies are briefly introduced. 1. Global topology: the Global topology is the most widely used in the literature. Each particle is directly connected to all other particles in the swarm and is each other's neighbors, so the particles can quickly exchange information, as illustrated in Figure 1. Thus, the algorithm with this topology can achieve convergence fast, but there is a risk of falling into a local optimum [16]. 2. Local topology: in the Local topology, the particles are directly connected to their \(m\) immediate neighbors. When \(m=2\), each particle only has two neighbors, as illustrated in Figure 2. Different regions in the search space can be simultaneously explored by using this topology [17]. Figure 1: Global topology. 3. Von Neumann topology: the Von-Neumann topology is a grid structure, as shown in Figure 3; each particle is connected to its four neighbors: top, bottom, left, and right. 4. Wheel topology: as shown in Figure 4, the particles using the Wheel topology are isolated from each other, and one particle is randomly selected as the focal point for all information flow. 5. Four Clusters topology: there are four subgroups in the four clusters topology, as displayed in Figure 5. The particles in each subgroup spread information under Global topology; each subgroup communicates with each other through three particles. Figure 4: Wheel topology. Figure 3: Von Neumann topology. Figure 2: Local topology. 6. Clan topology: the Clan topology is a type of dynamic topology that was proposed by [PERSON] et al. [17], in which the swarm is divided into several subgroups, called clans. The particles in each clan adjust their positions under the fully-connected structure (Global topology). Figure 6 shows an example of four clans (A, B, C, and D) used in this article. Each clan has five particles and the particle with the best fitness is selected as the leader of the clan in each iteration. In Figure 6, the leaders of A\({}_{3}\), B\({}_{5}\), C\({}_{5}\), and D\({}_{1}\) are marked in gray. Subsequently, a conference between the leaders occurs. In the conference, only the leaders of each clan are employed to perform a new PSO search. The leaders' members may change during the search process. When the conference takes place, it can use the Global or Local topologies to exchange information and is referred to as Clan Global topology and Clan Local topology. As shown in Figure 7, the Clan Global topology adopts the global information propagation mechanism among the leaders, enhancing the exploitation ability. In the Clan Local topology of Figure 8, the leaders take the local information propagation mechanism that can strengthen their exploration ability. Figure 5: Four Clusters topology. Figure 6: Individual clans. are \((k_{i-1},k_{i+1},k+1_{i})\), and \((k_{i-1},k_{i+1},k-1_{i})\), respectively. Moreover, the 'rotation skill' is added in this topology in order to reduce the possibility of falling into a local optimum. Thus, if the layer does not improve its own best location in \(t_{r}\) iterations, it will be rotated. A rotation example can be seen in Figure 10. It can be seen that the three layers of particles are named from **a** to **i**, and the particle **e** communicates with its neighbors \(\{\textbf{d},\textbf{f},\textbf{b},\textbf{h}\}\). After the rotation, its neighborhood is changed to \(\{\textbf{d},\textbf{f},\textbf{a},\textbf{g}\}\).