AUTHOR=Du Weiwei , Zhu Jian , Gao Haiping , Li Xianyong 

TITLE=Consensus analysis of the weighted corona networks

JOURNAL=Frontiers in Physics

VOLUME=10

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.948247

DOI=10.3389/fphy.2022.948247

ISSN=2296-424X

ABSTRACT=<p>The consensus of complex networks has attracted the attention of many scholars. The graph operation is a common method to construct complex networks, which is helpful in studying the consensus of complex networks. Based on the corona networks <italic>G</italic><sub>1</sub>◦<italic>G</italic><sub>2</sub>, this study gives different weights to the edges of <italic>G</italic><sub>1</sub>◦<italic>G</italic><sub>2</sub> to obtain the weighted corona networks <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>◦</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> and studies the consensus of <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>◦</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>. The consensus of the networks can be measured by coherence. First, the Laplacian polynomial of <inline-formula id="inf3"><mml:math id="m3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>◦</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> is derived by using the properties of an orthogonal matrix. Second, the relationship between the first-order coherence of <inline-formula id="inf4"><mml:math id="m4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>◦</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>̃</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula> and <italic>G</italic><sub>1</sub> is deduced by using the relevant properties of the determinant and the conclusion of polynomial coefficients and the principal minors of the matrix. Third, the join operation is introduced to further simplify the analytical formula of network coherence. Finally, a specific network example is used to verify the validity of the conclusion.</p>