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 =0. In the infinite network-size limit (N), we obtain a continuous transition with the density of activated edges growing like ^1 and with the diameter-expansion coefficient growing like ^2 in the regular network, and first-order transitions with discontinuous jumps in and at =0 for the small-world (SW) network and the Barab\'asi-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when NN₂, where the crossover size scales as N₂^-2 for the regular network, N₂ (^-2) for the SW network, and N₂ (|ln|^-2) for the SF network. In a transient regime with NN₂, there is an infinite-order transition with -/ (^2lnN) for the SW network and -/ (^2lnN/lnlnN) for the SF network. It shows that the transport pattern is practically most stable in the SF network.