  A complete understanding of real networks requires us to understand the
consequences of the uneven interaction strengths between a system's components.
Here we use the minimum spanning tree (MST) to explore the effect of weight
assignment and network topology on the organization of complex networks. We
find that if the weight distribution is correlated with the network topology,
the MSTs are either scale-free or exponential. In contrast, when the
correlations between weights and topology are absent, the MST degree
distribution is a power-law and independent of the weight distribution. These
results offer a systematic way to explore the impact of weak links on the
structure and integrity of complex networks.
