Analysis of the consensus of double-layer chain networks

The multi-layer network topology structures directly affect the robustness of network consensus. The different positions of edges between layers will lead to a great difference in the consensus of double-layer chain networks. Finding the optimal positions of edges for consensus can help to design the network topology structures with optimal robustness. In this paper, we first derive the coherence of double-layer chain networks with one and two connected edges between layers by graph theory. Secondly, the optimal and worst connection edges positions of the two types of networks are simulated. When there is one edge between layers, the optimal edge connection position is found at 1/2 of each chain, and the worst edge connection position is found at the end node of the chain. When there are two edges between layers, the optimal edges connection positions are located at 1/5 and 4/5 of each chain respectively, and the worst edges connection positions are located at the end node of the chain and its neighbor node. Furthermore, we find that the optimal edge connection positions are closely related to the number of single-layer network nodes, and obtain their specific rules.

Automated summary

The network topology structures of multi-layer networks have a direct impact on network consensus robustness. Different positions of edges between layers result in significant differences in consensus of double-layer chain networks. Finding optimal edge positions for consensus can aid in designing network topology structures with optimal robustness. This study derives the coherence of double-layer chain networks with one and two connected edges between layers using graph theory and conducts simulations to determine optimal and worst connection edge positions for the two types of networks. The optimal edge connection position is found at 1/2 of each chain when there is one edge between layers and at 1/5 and 4/5 of each chain when there are two edges between layers. The worst edge connection position is found at the end node of the chain and its neighbor node. Furthermore, the study reveals a close relationship between the optimal edge connection positions and the number of nodes in the single-layer network, providing specific rules.