Minimum Weight Dynamo and Fast Opinion Spreading

We consider the following multi--level opinion spreading model on networks. Initially, each node gets a weight from the set $\{0,\ldots,k-1\}$, where such a weight stands for the individuals conviction of a new idea or product. Then, by proceeding to rounds, each node updates its weight according to the weights of its neighbors. We are interested in the initial assignments of weights leading each node to get the value $k-1$ --e.g. unanimous maximum level acceptance-- within a given number of rounds. We determine lower bounds on the sum of the initial weights of the nodes under the irreversible simple majority rules, where a node increases its weight if and only if the majority of its neighbors have a weight that is higher than its own one. Moreover, we provide constructive tight upper bounds for some class of regular topologies: rings, tori, and cliques.