  A set of identical, mobile agents is deployed in a weighted network. Each
agent has a battery -- a power source allowing it to move along network edges.
An agent uses its battery proportionally to the distance traveled. We consider
two tasks : convergecast, in which at the beginning, each agent has some
initial piece of information, and information of all agents has to be collected
by some agent; and broadcast in which information of one specified agent has to
be made available to all other agents. In both tasks, the agents exchange the
currently possessed information when they meet. The objective of this paper is
to investigate what is the minimal value of power, initially available to all
agents, so that convergecast or broadcast can be achieved. We study this
question in the centralized and the distributed settings. In the centralized
setting, there is a central monitor that schedules the moves of all agents. In
the distributed setting every agent has to perform an algorithm being unaware
of the network. In the centralized setting, we give a linear-time algorithm to
compute the optimal battery power and the strategy using it, both for
convergecast and for broadcast, when agents are on the line. We also show that
finding the optimal battery power for convergecast or for broadcast is NP-hard
for the class of trees. On the other hand, we give a polynomial algorithm that
finds a 2-approximation for convergecast and a 4-approximation for broadcast,
for arbitrary graphs. In the distributed setting, we give a 2-competitive
algorithm for convergecast in trees and a 4-competitive algorithm for broadcast
in trees. The competitive ratio of 2 is proved to be the best for the problem
of convergecast, even if we only consider line networks. Indeed, we show that
there is no (2 -- $\epsilon$)-competitive algorithm for convergecast or for
broadcast in the class of lines, for any $\epsilon$ \textgreater{} 0.
