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 $\varepsilon =0.\mathrm{}$ In the infinite network-size limit $\left(\overrightarrow{N}\infty \right),$ we obtain a continuous transition with the density of activated edges $\Phi$ growing like $\Phi \sim {\varepsilon }^1$ and with the diameter-expansion coefficient ${\rm Y}$ growing like ${\rm Y}\sim {\varepsilon }^2$ in the regular network, and first-order transitions with discontinuous jumps in $\Phi$ and ${\rm Y}$ at $\epsilon =0\mathrm{}$ for the small-world (SW) network and the Barabási-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when $N\gg N_c,$ where the crossover size scales as $N_c\sim {\varepsilon }^{-2}$ for the regular network, $N_c\sim \exp \left(\alpha {\varepsilon }^{-2}\right)$ for the SW network, and $N_c\sim \exp \left(\alpha |\ln \varepsilon |{\varepsilon }^{-2}\right)$ for the SF network. In a transient regime with $N\ll N_c,$ there is an infinite-order transition with $\Phi \sim {\rm Y}\sim \exp [-\alpha /({\varepsilon }^2\ln N\left)\right]$ for the SW network and $\sim \exp [-\alpha /({\varepsilon }^2\ln N/\ln \ln N\left)\right]$ for the SF network. It shows that the transport pattern is practically most stable in the SF network.

• Received 22 August 2002

DOI:https://doi.org/10.1103/PhysRevE.66.066127