Time and Energy Costs for Intra-Layer Consensus of Multi-Layer Networks

Multi-layer network representation has been shown to well characterize real networks, and has revealed some potential phenomenon beyond the interpretation ability of single-layer network successfully. Despite its uniqueness and excitement, intra-layer consensus gets less attention. This paper proposes a discontinuous protocol to address the finite-time intra-layer consensus problem for multi-layer networks. Different from the traditional results of finite-time consensus schemes work only on undirected or detail-balanced networks, a more general case that the network contains a directed spanning tree is considered with a discontinuous protocol. On the other hand, without limiting networks to be of drive-response type or specifically duplexed, the inter-layer topology is assumed to be unidirectional. In addition, both time and energy costs for achieving consensus are studied. Finally, an example is given to verify the theoretical results.