First- and second-order network coherence in N-duplication weighted corona networks

This paper studies first- and second-order coherence problems for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math> -duplication weighted corona networks subject to stochastic disturbances. Explicit expressions of the coherence for first-order (and second-order) dynamics, which are determined by the sum of the reciprocal (and square of reciprocal) of each nonzero eigenvalue of the Laplacian matrix, are derived. In particular, for both first- and second-order systems, the analytical formulas of the network coherence are presented from two different perspectives. Based on these formulas, the influence of the duplication <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math> , the weight <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mi>ω</mml:mi></mml:mrow></mml:math> , and the factor networks <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> on the network coherence of the corona network <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>◦</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> is investigated. Some noteworthy topological properties of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m8"><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math> -duplication weighted corona network are also revealed.