Design and Control of Swarm Dynamics by Roland Bouffanais

Author:Roland Bouffanais
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

The very first characterization of the SSN pertains to its connectedness, which, in a k-nearest graph representing the topological interactions (see Sect. 3.​2.​2), heavily depends on the value of the outdegree k (Fig. 4.3d, e). The existence of a critical value, , for the outdegree k such that for the k-nearest graph is connected, has never been proved. However, Balister et al. [20] proved the existence of in the probabilistic sense. More specifically, they proved that for

(4.1)

where N is the number of nodes—i.e., the number of swarming agents—the probability for any randomly generated k-nearest graph to be connected tends to one. In Eq. (4.1), c is a constant and the smallest value found so far is 0.9967 [20]. It is important keeping in mind that those mathematical results were obtained under the assumption that N is large. When collective motion is considered, the number of agents considered ranges from dozens to a few thousands, and rarely more [8]. It is therefore important to assess numerically the validity of Eq. (4.1) for values of N smaller than 1,000. Figure 4.4 shows that even for small values of N, continues to scale linearly with on average. Moreover, the average value of the coefficient c here is found equal to 1.15—this value decreases with increasing N, which is consistent with the value 0.9967 found in Ref. [20] for large N.

The study of the connectedness of the SSN uncovers the existence of a relationship between swarm size N and the number k of nearest neighbors influencing any agent’s behavior. Indeed, general results from graph theory applied to the study of the SSN connectedness take a particular significance in the context of dynamic collective behavior where N may not necessarily be very large and k, cannot possibly exceed at most 15 to 20 due to the intrinsic bandwidth limitations in signaling, sensing and internal information processing. To better appreciate these results, the dependence of the probability of connectedness of the SSN as a function of N for different values of k is depicted in Fig. 4.5, which reveals the profound relationship between connectedness of the swarm and the number of agents N, for different values of the outdegree k. This important result was already suggested by Eq. (4.1) and is related to the concept of percolation at critical connectivity. As was seen in Sect. 3.​4.​1, at the critical point, system-level communications become possible and information can flow between agents throughout the entire swarm, and in a very effective manner. In addition, with adaptive networks, there is a feedback of global information into the topological evolution through the local agent’s dynamics that can allow the SSN to self-organize toward a critical state.

Fig. 4.5Probability of connectedness for the SSN versus number of agents N for different values of the number of nearest neighbors k. The SSN corresponds to a specific configuration of the swarm in which N nodes are placed in a unit square independently through a uniform distribution. Then each node is connected to its k nearest neighbors to form the k-nearest graph.

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