Achieving consensus on networks with antagonistic interactions

Consensus protocols achieve an agreement among agents thanks to the collaborative efforts of all agents, expresses by a (connected) communication graph with nonnegative weights. The question we ask in this paper is the following: is it possible to achieve a form of agreement in presence of antagonistic interactions, modeled as negative weights on the communication graph? The answer to this question is affirmative: on signed networks all agents can converge to a consensus value which is the same for all except for the sign. Necessary and sufficient conditions are obtained to describe the cases when this is possible.