Before revision:  In this setting we prove two phase transition phenomena, both occurring with high probability, depending on the bias p and on the initial unbalance toward state \sigma. More in detail,  we prove that for every k \geq 3 \geq3 there exists a p_k^\star such that if p>p _k^\star the process reaches in O(1) rounds a \sigma-almost-consensus, i.e., a configuration where a fraction 1-o(1) of the volume is in state \sigma.
After revision:  In this setting we prove two phase transition phenomena, both occurring with high probability, depending on the bias p and on the same critical value. By considering initial configurations in which each node has probability q \in (\frac{1 we prove that for every k \geq 3 \geq3 there exists a p_k^\star such that if p>p _k^\star the process reaches in O(1) rounds a \sigma-almost-consensus, i.e., a configuration where a fraction 1-o(1) of the volume is in state \sigma.