The Coherence and Robustness Analysis for a Family of Unbalanced Networks

This study focuses on the coherence in a class of unbalanced networks, emphasizing the investigation of the impact of leader selection and network parameters on coherence. We discuss three different categories of consensus algorithms: leaderless algorithms of first and second orders, and first-order leader-follower algorithm. Firstly, we derive exact solutions for leaderless network coherence in the first and second orders. Secondly, we design various leader allocation schemes for the first-order leader-follower networks, obtaining expressions correlated with the network scale <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula>, the double-ring cardinalities <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P$</tex-math></inline-formula> and q, and the number of leaders <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula>. Thirdly, we conduct numerical simulations and robustness analyses to explore the impact of leader quantity and topology structure on network coherence. Specifically, the conclusions are as follows: (1) When N decreases, an increase in k demonstrates improved network coherence. (2) The smaller the difference between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q$</tex-math></inline-formula>, the better the coherence. (3) The smaller the sum of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P$</tex-math></inline-formula>, the stronger the robustness. (4) Uniform distribution of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K_1$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K_2$</tex-math></inline-formula> leads to optimal coherence. Finally, the Laplacian energy and Kirchhoff index of the network are analyzed.