Consensus over Networks with Merging Agents

We study consensus among agents that physically merge when they come close. The protocol is a hybrid system. During flows, the agents run the consensus protocol over the current graph. A jump occurs when two agents that are not yet merged come within a docking distance: the states of their two clusters are reset to their common center of mass (state merge), and every agent of the new cluster is connected to the other agents of the cluster and to all of their neighbors (neighborhood merge). Merged agents thus have equal states and identical neighborhoods, so the flow keeps them together. We show that the resulting graph is a blow-up of a contraction of the initial graph, in which each cluster is contracted to a node and then expanded back into a clique. The network therefore evolves as a consensus network over the contracted graph with cluster masses, and the protocol needs no sensing links beyond the original ones. Using this structure, we show that all agents merge at the initial centroid in finite time, with an explicit bound that holds for every order of the merges. Extensions to formations, collision avoidance, and leader--follower networks are given, and the results are illustrated by modular self-assembly and cooperative transport.