The xylem is one of the two long distance transport tissues in plants, providing a low resistance pathway for water movement from roots to leaves. Its properties determine how much water can be transported and transpired and, at the same time, the plant's vulnerability to transport dysfunctions (the formation and propagation of emboli) associated to important stress factors, such as droughts and frost. Both maximum transport efficiency and safety against embolism have classically been attributed to the properties of individual conduits or of the pit membrane connecting them. But this approach overlooks the fact that the conduits of the xylem constitute a network. The topology of this network is likely to affect its overall transport properties, as well as the propagation of embolism through the xylem, since, according to the air-seeding hypothesis, drought-induced embolism propagates as a contact process (i.e., between neighbouring conduits). Here we present a model of the xylem that takes into account its system-level properties, including the connectivity of the xylem network. With the tools of graph theory and assuming steady state and Darcy's flow we calculated the hydraulic conductivity of idealized wood segments at different water potentials. A Monte Carlo approach was adopted, varying the anatomical and topological properties of the segments within biologically reasonable ranges, based on data available from the literature. Our results showed that maximum hydraulic conductivity and vulnerability to embolism increase with the connectivity of the xylem network. This can be explained by the fact that connectivity determines the fraction of all the potential paths or conduits actually available for water transport and spread of embolism. It is concluded that the xylem can no longer be interpreted as the mere sum of its conduits, because the spatial arrangement of those conduits in the xylem network influences the main functional properties of this tissue. This brings new arguments into the long-standing discussion on the efficiency vs. safety trade-off in the plants' xylem.