## 4 Analysis of the Properties of LCPSO
### Symmetry and Robustness of the Swarm Formation  
Due to the nature of the potential functions \(P^{a}(d_{i,j})\) and \(P^{r}(d_{i,j})\), their gradient is _odd_. Consequently, there is a reciprocity of the interactions between agents, with respect to the origin [39]. These reciprocal interactions will naturally lead the swarm to have a _symmetrical_ formation with respect to the center of gravity of the swarm \(\mathbf{\tilde{p}}\) when it is stable in the sense of Definition 2[39]. Figure 2: Possible states for the swarm; the target is in red. (**a**) Steady state; (**b**) Transition state. Contrary to the swarm models detailed in [38; 39], all interactions are not bidirectional when looking at the whole system. Indeed, if this is true for interactions of repulsion, attraction relationships are unidirectional and directed towards the attractor. One could therefore assume that a change of attractor influences the strength of the formation when it is stable in the sense of Definition 1. However, with the LCPSO Equation (18), the formation is robust to this, regardless of the dimension \(D\). By assuming \(N\geq 2\) agents using the flocking speed vector described in (18), whatever the attractor \(\mathbf{p}^{g}(t)\), the equilibrium distance of agent \(i\) with the other agents will always be the same. The proof is in the Appendix A. With the help of Theorem 2 from [39], we can see that the formation is bounded whatever the dimension \(D\):  
If the agents follow the LCPSO Equation (18), as time progresses, all the members of the swarm will converge to a hyperball:  
\[B_{\epsilon}(\bar{\mathbf{p}})=\{\mathbf{p}:\|\mathbf{p}-\bar{\mathbf{p}}\| \leq\epsilon\}\quad\text{where }\epsilon=\frac{r_{eq}N}{2}\]  
The proof is not present in this paper, because it is too similar to that of Theorem 2 from [39]. \(\epsilon\) increases linearly as a function of \(N\) in Theorem 5. We can see in Figure 2(b) that the evolution of \(r_{max}\) as a function of \(N\), in Dimension 1 or 2, does not have a linear evolution, but tends to \"flatten\" when \(N\) increases: thus, this bound is real, but not adapted to approach the size of the swarm in reality. Now, we will look more prospectively at the properties of the stable formation. We thus present conjectures, supported by simulation results, which will remain to be proven mathematically afterwards. We do not display results in Dimension 3, because the remarks would be redundant with those in Dimension 2. We use \(r_{max}\) and \(\rho(N)\) to illustrate the evolution of the stable formations when \(N\) increases, depending on the dimension. The results are shown in Figure 3. In dimension 1, the formation of the agents when the swarm is stable in the sense of Definition 2 is _unique_, so we do not need several samples. In higher dimension, the multiplicity of emergent formations as a function of \(N\) lead up to several possible formations. In dimension 1, we can see in Figure 2(a) that the more \(N\) increases, the more the swarm spacing \(\rho(N)\) decreases; this is rather logical, because the surface taken by the swarm widens very quickly, as we can see in Figure 2(b). Hence, the more \(N\) increases, the more certain agents are distant from the attractor, the more the attraction strength is high, the more the swarm is compact. In Dimension 2, when \(N=3\), the equilibrium formation is an equilateral triangle. When \(N\) is higher, the possible formations approach a circle whose the center of gravity determines a point of symmetry in the interactions between agents. This formation presents one or several layers where the position of the agents is aligned on circles, and it becomes more difficult to predict it, as shown in Figure 2(f), with 2 layers for \(N=15\) agents. Moreover, in Figure 2(b), we have \(1\leq\rho(N)\leq 1.3\) whatever \(N\) and our samples: our parameter \(r_{eq}\) is a good representation of swarm spacing.