| Summary: | 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>.
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