Robustness of Consensus of Two-Layer Ring Networks
The topology structure of multi-layer networks is highly correlated with the robustness of consensus. This paper investigates the influence of different interlayer edge connection patterns on the consensus of the two-layer ring networks. Two types of two-layer ring network models are first considere...
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MDPI AG
2023-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/5/1085 |
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| author | Zhijun Li Haiping Gao Zhiyong Shang Wenming Zhang |
| author_facet | Zhijun Li Haiping Gao Zhiyong Shang Wenming Zhang |
| author_sort | Zhijun Li |
| collection | DOAJ |
| description | The topology structure of multi-layer networks is highly correlated with the robustness of consensus. This paper investigates the influence of different interlayer edge connection patterns on the consensus of the two-layer ring networks. Two types of two-layer ring network models are first considered: one is a kind of two-layer ring network with two linked edges between layers (Networks-a), and the other is a kind of two-layer ring network with three linked edges between layers (Networks-b). Using the Laplacian spectrum, the consensus of the network model is derived. The simulation experiments are used to demonstrate the influence of different interlayer edge connection patterns on the consensus of networks. To determine the best edge connection pattern for Networks-a and Networks-b, the number of nodes in a single-layer ring network is denoted by <i>n</i>. The best edge connection pattern for Networks-a is 1 & <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>. Furthermore, <i>n</i> is subdivided into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mi>k</mi><mo>,</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula>, and the best edge connection patterns of Networks-b are near 1 & <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> & <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T03:16:25Z |
| format | Article |
| id | doaj.art-7496d90e346843a28de038ac0be1b96b |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T03:16:25Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
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| series | Symmetry |
| spelling | doaj.art-7496d90e346843a28de038ac0be1b96b2023-11-18T03:30:43ZengMDPI AGSymmetry2073-89942023-05-01155108510.3390/sym15051085Robustness of Consensus of Two-Layer Ring NetworksZhijun Li0Haiping Gao1Zhiyong Shang2Wenming Zhang3Academic Affairs Office, Xinjiang Institute of Light Industry Technology, Urumqi 830023, ChinaDepartment of Basic Science, Xinjiang Institute of Light Industry Technology, Urumqi 830023, ChinaMechanical Engineering Laboratory, Xinjiang Institute of Engineering, Urumqi 830023, ChinaDepartment of Information and Software, Xinjiang Institute of Light Industry Technology, Urumqi 830023, ChinaThe topology structure of multi-layer networks is highly correlated with the robustness of consensus. This paper investigates the influence of different interlayer edge connection patterns on the consensus of the two-layer ring networks. Two types of two-layer ring network models are first considered: one is a kind of two-layer ring network with two linked edges between layers (Networks-a), and the other is a kind of two-layer ring network with three linked edges between layers (Networks-b). Using the Laplacian spectrum, the consensus of the network model is derived. The simulation experiments are used to demonstrate the influence of different interlayer edge connection patterns on the consensus of networks. To determine the best edge connection pattern for Networks-a and Networks-b, the number of nodes in a single-layer ring network is denoted by <i>n</i>. The best edge connection pattern for Networks-a is 1 & <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>. Furthermore, <i>n</i> is subdivided into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mi>k</mi><mo>,</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula>, and the best edge connection patterns of Networks-b are near 1 & <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> & <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2073-8994/15/5/1085multi-layer ring networktopology structureconsensusrobustnessedge connection patterns |
| spellingShingle | Zhijun Li Haiping Gao Zhiyong Shang Wenming Zhang Robustness of Consensus of Two-Layer Ring Networks Symmetry multi-layer ring network topology structure consensus robustness edge connection patterns |
| title | Robustness of Consensus of Two-Layer Ring Networks |
| title_full | Robustness of Consensus of Two-Layer Ring Networks |
| title_fullStr | Robustness of Consensus of Two-Layer Ring Networks |
| title_full_unstemmed | Robustness of Consensus of Two-Layer Ring Networks |
| title_short | Robustness of Consensus of Two-Layer Ring Networks |
| title_sort | robustness of consensus of two layer ring networks |
| topic | multi-layer ring network topology structure consensus robustness edge connection patterns |
| url | https://www.mdpi.com/2073-8994/15/5/1085 |
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