The paper proposes new consensus protocols for the agreement problem in networks of agents with a discrete time model. A new class of consensus algorithms is introduced on the basis of the positive splitting of the standard iteration matrix. In the framework of non negative matrix theory, some results are proved to guarantee the convergence of the proposed algorithms. In addition, numerous numerical experiments show that the proposed iterative schemes enjoy good rate of convergence even in the cases in which the standard iterative algorithms do not guarantee good performances.