Multi-level dynamo and opinion spreading

We consider the following multi-level opinion spreading model on networks. Initially, each node gets a weight, from the set {0,.,k-1}, which measures the individual conviction of a new idea or product. Then, by proceeding in rounds, each node updates its weight according to those of its neighbours. We study k-dynamos that are initial assignments of weights leading each node to get the value k-1-e.g. unanimous maximum level of acceptance-within a given number of rounds; the goal is to minimize the sum of the initial weights of the nodes. We determine lower bounds on the sum of the initial weights under the irreversible simple majority rules, where a node increases its weight if and only if the majority of its neighbours have a weight that is higher than its own. We study the relations among 2-dynamos and k-dynamos, with and without a bound on the number of rounds needed to reach the desired all-(k-1) configuration. Moreover, we provide constructive tight upper bounds for some classes of regular topologies: rings, tori and cliques.