Distributed consensus with multi-equilibria in directed networks

Topology design for group consensus in directed multi-agent systems

“…However, for the secondary layer subgraphs, there is a unique root that receives information from the rest of the graph which we call the principal root of the subgraph and denote by r Gs,j . Note here that with the above definition, each vertex in G will either be in a primary or a secondary layer subgraph and these subgraphs can be uniquely determined [9,10]. Note also that an isolated vertex is a primary layer subgraph itself by definition.…”

“…Remark 2.10. In previous studies, it is shown that the number of groups formed in a multi-agent network with agent dynamics (1) or (2) is defined as K = l p + l s , i. e., the sum of the number of primary and secondary layer subgraphs [9,10]. One of the main objectives of this paper is to design an algorithm that can be used to generate a directed graph with the desired numbers of primary and secondary layer subgraphs, and the desired numbers of agents in each subgraph.…”

Optimizing Pinned Nodes to Maximize the Convergence Rate of Multiagent Systems with Digraph Topologies

“…Network without Spanning Trees. Without loss of generality, we suppose that has the following form [26]…”

“…, 1 is a Laplacian matrix and its corresponding subgraph has a spanning tree (isolated node can be seen as a subgraph that has a spanning tree with itself being the root). In [26], the subgraphs corresponding to , = 1, . .…”

“…The pinned node number is 6 in this simulation. By the primary layer detection algorithm in [26], we find that there are 4 primary layer subgraphs in 5 . Denote the root set of these 4 primary layer subgraphs by S , = 1, 2, 3, 4.…”