In this work we extend the population protocol model of Angluin et al., in order to model more powerful networks of very small resource limited artefacts (agents) that is possible to follow some unpredictable passive movement. These agents communicate in pairs according to the commands of an adversary scheduler. A directed (or undirected) communication graph encodes the following information: each edge (u,õ) denotes that during the computation it is possible for an interaction between u and õ to happen in which u is the initiator and õ the responder. The new characteristic of the proposed mediated population protocol model is the existance of a passive communication provider that we call mediator. The mediator is a simple database with communication capabilities. Its main purpose is to maintain the permissible interactions in communication classes, whose number is constant and independent of the population size. For this reason we assume that each agent has a unique identifier for whose existence the agent itself is not informed and thus cannot store it in its working memory. When two agents are about to interact they send their ids to the mediator. The mediator searches for that ordered pair in its database and if it exists in some communication class it sends back to the agents the state corresponding to that class. If this interaction is not permitted to the agents, or, in other words, if this specific pair does not exist in the database, the agents are informed to abord the interaction. Note that in this manner for the first time we obtain some control on the safety of the network and moreover the mediator provides us at any time with the network topology. Equivalently, we can model the mediator by communication links that are capable of keeping states from a edge state set of constant cardinality. This alternative way of thinking of the new model has many advantages concerning the formal modeling and the design of protocols, since it enables us to abstract away the implementation details of the mediator. Moreover, we extend further the new model by allowing the edges to keep readable only costs, whose values also belong to a constant size set. We then allow the protocol rules for pairwise interactions to modify the corresponding edge state by also taking into account the costs. Thus, our protocol descriptions are still independent of the population size and do not use agent ids, i.e. they preserve scalability, uniformity and anonymity. The proposed Mediated Population Protocols (MPP) can stably compute graph properties of the communication graph. We show this for the properties of maximal matchings (in undirected communication graphs), also for finding the transitive closure of directed graphs and for finding all edges of small cost. We demonstrate that our mediated protocols are stronger than the classical population protocols. First of all we notice an obvious fact: the classical model is a special case of the new model, that is, the new model can compute at least the same things with the classical one. We then present a mediated protocol that stably computes the product of two nonnegative integers in the case where G is complete directed and connected. Such kind of predicates are not semilinear and it has been proven that classical population protocols in complete graphs can compute precisely the semilinear predicates, thus in this manner we show that there is at least one predicate that our model computes and which the classical model cannot compute. To show this fact, we state and prove a general Theorem about the composition of two mediated population protocols, where the first one has stabilizing inputs. We also show that all predicates stably computable in our model are (non-uniformly) in the class NSPACE(m), where m is the number of edges of the communication graph. Finally, we define Randomized MPP and show that, any Peano predicate accepted by a Randomized MPP, can be verified in deterministic polynomial time.