Abstract
A key function of mobile networks is the ability to dynamically reshape itself to any desired geometry. Lacking absolute position awareness, agents often rely on distance-limited inter-agent spatial measurements to maintain state awareness. Methods of formation control must therefore ensure a minimal level of persistent pairwise measurement feedback throughout transition, giving rise to the classic connectivity maintenance problem. To address this problem, we propose a method of structure-preserving assignment, matching agents to desired positions such that persistent global connectivity is naturally and automatically satisfied under smooth transition. Compared to other approaches, this complementary technique reduces reliance on aggressive or costly mid-flight formation control protocols. The technique is shown to scale and even improve with network size.
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This work was supported by U.S. DEVCOM Army Research Laboratory.
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Appendix A Graph Density
Appendix A Graph Density
In the model described in Section 3.2, the distribution of occupation spectra is formulated as a function of occupation number, \( \eta \). It is not immediately obvious, however, how \( \eta \) relates to more conventional graph metrics. Intuitively, we expect \( \eta \) will increase with graph density, but we would like to make this statement more precise. Graph density of undirected simple graphs (i.e., without multi-edges) is often defined to be the ratio of the number of edges with respect to the maximum possible edges [30], \( \rho = \vert E \vert /\left( {\begin{array}{c}n\\ 2\end{array}}\right) \). It will also be convenient to reference the fractional occupation \( \phi = \frac{\eta }{k(n)} \) to normalize and compare results across different graph orders n.
Figure 18 shows the conditional probability distribution \( \textrm{Pr}(\phi \ge \phi _0 \mid \rho ) \) for \( n=7 \). Note that the fractional occupation rises monotonically with graph density. The data is generated from numerical analysis of 60 thousand random connected simple graphs on \( n=7 \) vertices. Graphs are drawn uniformly from the Erdös-Rényi model with densities \( \rho \in [\frac{n-1}{\left( {\begin{array}{c}n\\ 2\end{array}}\right) }, \frac{3}{4}] \). Note that \( \frac{n-1}{\left( {\begin{array}{c}n\\ 2\end{array}}\right) } \) is the minimum threshold density for a graph to be connected. This can be seen from the fact that a connected graph with \( n-1 \) edges is itself an nth order tree and will accommodate exactly one occupied class (since no single spanning tree can belong to two classes). Fewer than \( n-1 \) edges will not support any connected spanning tree and is necessarily disconnected.
This n-normalized distribution and monotonic dependence is similar for \( n=9 \), as shown in Fig. 19. This mapping between occupation number and graph density serves to express results of Section 4 in terms of conventional graph density and provides a means of extrapolating requirements for practical network applications.
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Hamaoui, M. Connectivity Maintenance through Unlabeled Spanning Tree Matching. J Intell Robot Syst 110, 15 (2024). https://doi.org/10.1007/s10846-024-02048-9
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DOI: https://doi.org/10.1007/s10846-024-02048-9