Mitigation of complex behavior over networked systems: analysis of spatially invariant structures.

*(English)*Zbl 1360.93059Summary: In this paper, we consider a simple distributed averaging system, which incorporates various communication constraints including delays, noise, and link failures. It has been shown by the authors [“Distributed averaging under constraints on information exchange: emergence of Lévy flights”, IEEE Trans. Autom. Control 57, No. 10, 2435–2449 (2012; doi:10.1109/TAC.2012.2186093)], that such networked system generates a collective Lévy flight behavior when part of the system loses Mean Square (MS) stability. We focus on spatially invariant architectures to gain more insights into how model parameters affect emergence of this complex scale-invariant behavior, and to seek structures robust to communication constraints. Specifically, we develop a computational expression for checking MS stability, which is scalable with the number of unreliable links. We derive the closed form formulas from this expression in the limiting case of zero and large delays, and in the case of large number of nodes. In the limit of large delays, we derive various results that are independent of the network size and its specific interconnections. We find that small inter-agent coupling improves the robustness of the system. Networks with larger connectivity tend to be more fragile in the presence of fading connections for fixed inter-agent coupling. That gossiping improves the robustness and that the lattice is the most robust among the spatially invariant systems with generalized circulant interconnections.

##### MSC:

93A14 | Decentralized systems |

93A15 | Large-scale systems |

93E15 | Stochastic stability in control theory |

##### Keywords:

networked systems; distributed averaging; consensus; mean square stability; Lévy flights; hyper-jump diffusion; scale invariance; complex systems##### Software:

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\textit{J. Wang} and \textit{N. Elia}, Automatica 49, No. 6, 1626--1638 (2013; Zbl 1360.93059)

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