text: abstract—the paper studies the problem of achieving consensus in multi-agent systems in the case
  where the dependency digraph γ has no spanning in-tree. we consider the regularization protocol that
  amounts to the addition of a dummy agent (hub) uniformly connected to the agents. the presence
  of such a hub guarantees the achievement of an asymptotic consensus. for the “evaporation” of the
  dummy agent, the strength of its influences on the other agents vanishes, which leads to the concept
  of latent consensus. we obtain a closed-form expression for the consensus when the connections of
  the hub are symmetric; in this case, the impact of the hub upon the consensus remains fixed. on
  the other hand, if the hub is essentially influenced by the agents, whereas its influence on them tends
  to zero, then the consensus is expressed by the scalar product of the vector of column means of the
  laplacian eigenprojection of γ and the initial state vector of the system. another protocol, which
  assumes the presence of vanishingly weak uniform background links between the agents, leads to the
  same latent consensus.
  keywords: consensus, multi-agent system, decentralized control, regularization, eigenprojection, degroot’s iterative pooling, pagerank, laplacian matrix of a digraph.