Sparse solutions to the average consensus problem via l1-norm regularization of the fastest mixing Markov-chain problem

In the “consensus problem” on multi-agent systems, in which the states of the agents are “opinions”, the agents aim at reaching a common opinion (or “consensus state”) through local exchange of information. An important design problem is to choose the degree of interconnection of the subsystems so as to achieve a good trade-off between a small number of interconnections and a fast convergence to the consensus state, which is the average of the initial opinions under mild conditions. This paper addresses this problem through l₁-norm regularized versions of the well-known fastest mixing Markov-chain problem, which are investigated theoretically. In particular, it is shown that such versions can be interpreted as “robust” forms of the fastest mixing Markov-chain problem. Theoretical results useful to guide the choice of the regularization parameters are also provided, together with a numerical example.