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The task management is a key point in grid applications and can highly influence their efficiency. There are many solutions that we can classify according to their centralization degree: fully centralized, semi-distributed and fully distributed. A fully distributed approach leaves the nodes in charge of the scheduling: the schedule is performed thanks to the local view of the system. Such a local treatment reduces the communication cost but it must not impact on the computational power devoted to applications. This kind of solution seems to be more scalable and flexible than a centralized one, especially in a highly volatile environment like peer-to-peer networks. In this article, we propose two fully distributed solutions for the task management based on random walks. We choose to maximize the computational power of nodes by reducing the maintenance cost of an underling structure (a spanning structure). So, it reduces the number of control message exchanges that is a critical point in networks with a low bandwidth or in which the energy of nodes must be saved. We analyze two methods called passive and active and we present some simulation results.