Abstract
Consensus protocol analysis suffers from unrealistic instantaneous information communication assumption. To address this, we shall consider arbitrarily large delays in consensus protocols and explore its impact on dynamical behavior of convergence. We formulate a conjecture about emergence of slowly oscillating periodic (SOP) orbits as delay becomes significant from sine-like waves to square-waves. Further, dynamical stability behavior of (i) minimally connected multi-agent configuration, i.e., a chain, (ii) fully connected graph, and finally the intermediate configuration, (iii) multiple agents connected in a circular fashion with exactly one full length closed path will be reported. It will be shown that for chain configuration, period doubling behavior will be observed irrespective of number of agents. For a fully connected network configuration guranteed and stable convergence to consensus point with equal contributions from all agents, i.e., peer-setup will be proved irrespective of delay and number of agents. Surprisingly, it will be demonstrated that multiple agents connected in a full length closed path will converge to consensus with equal contribution from all nodes irrespective of delay if number of agents is odd, while they will oscillate in the period doubling fashion if number of agents is even. A bifurcation diagram with link strength parameter which transitions a chain into fully connected closed path will be presented to illustrate interesting branching processing from period doubling to single equilibrium consensus point. Connection with theory of circulant matrix [11, 12] will be pointed out for future deeper investigations.
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Acknowledgements
Author would like to acknowledge deep gratitude toward Prof. A. F. Ivanov and Prof. A. N. Sharkovsky from whom he has learnt a lot. He is also grateful to researchers in circulant matrix [11, 12], FFT-based methods for diagonalizing circulant matrices and tridigonal matrix research communities whose work is directly relevant in solving consensus problems in circular/ring and in linear chain configurations due to their permutation structure.
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Ranjan, P. (2022). Instabilities of Consensus. In: Sarma, H.K.D., Balas, V.E., Bhuyan, B., Dutta, N. (eds) Contemporary Issues in Communication, Cloud and Big Data Analytics. Lecture Notes in Networks and Systems, vol 281. Springer, Singapore. https://doi.org/10.1007/978-981-16-4244-9_9
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DOI: https://doi.org/10.1007/978-981-16-4244-9_9
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