Stability of shortest paths in complex networks with random edge weights

We study shortest paths and spanning trees of complex networks with random edge weights. Edges which do not belong to the spanning tree are inactive in a transport process within the network. The introduction of quenched disorder modifies the spanning tree such that some edges are activated and the network diameter is increased. With analytic random-walk mappings and numerical analysis, we find that the spanning tree is unstable to the introduction of disorder and displays a phase-transitionlike behavior at zero disorder strength [formula presented] In the infinite network-size limit [formula presented] we obtain a continuous transition with the density of activated edges [formula presented] growing like [formula presented] and with the diameter-expansion coefficient [formula presented] growing like [formula presented] in the regular network, and first-order transitions with discontinuous jumps in [formula presented] and [formula presented] at [formula presented] for the small-world (SW) network and the Barabási-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when [formula presented] where the crossover size scales as [formula presented] for the regular network, [formula presented] for the SW network, and [formula presented] for the SF network. In a transient regime with [formula presented] there is an infinite-order transition with [formula presented] for the SW network and [formula presented] for the SF network. It shows that the transport pattern is practically most stable in the SF network.