  This work shows potentials for rapid self-organisation of sensor networks
where nodes collaborate to relay messages to a common data collecting unit
(sink node). The study problem is, in the sense of graph theory, to find a
shortest path tree spanning a weighted graph. This is a well-studied problem
where for example Dijkstra's algorithm provides a solution for non-negative
edge weights. The present contribution shows by simulation examples that simple
modifications of known distributed approaches here can provide significant
improvements in performance. Phase transition phenomena, which are known to
take place in networks close to percolation thresholds, may explain these
observations. An initial method, which here serves as reference, assumes the
sink node starts organisation of the network (tree) by transmitting a control
message advertising its availability for its neighbours. These neighbours then
advertise their current cost estimate for routing a message to the sink. A node
which in this way receives a message implying an improved route to the sink,
advertises its new finding and remembers which neighbouring node the message
came from. This activity proceeds until there are no more improvements to
advertise to neighbours. The result is a tree network for cost effective
transmission of messages to the sink (root). This distributed approach has
potential for simple improvements which are of interest when minimisation of
storage and communication of network information are a concern. Fast
organisation of the network takes place when the number $k$ of connections for
each node ({\em degree}) is close above its critical value for global network
percolation and at the same time there is a threshold for the nodes to decide
to advertise network route updates.
