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Nature is organized into complex, dynamical networks of species and their interactions, which may influence diversity and stability. However, network research is, generally, short-term and depict ecological networks as static structures only, devoid of any dynamics. This hampers our understanding of how nature responds to larger disturbances such as changes in climate. In order to remedy this we studied the long-term (12-yrs) dynamics of a flower-visitation network, consisting of flower-visiting butterflies and their nectar plants. Global network properties, i.e. numbers of species and links, as well as connectance, were temporally stable, whereas most species and links showed a strong temporal dynamics. However, species of butterflies and plants varied bimodally in their temporal persistance: Sporadic species, being present only 1-2(-5) years, and stable species, being present (9-)11-12 years, dominated the networks. Temporal persistence and linkage level of species, i.e. number of links to other species, made up two groups of species: Specialists with a highly variable temporal persistence, and temporally stable species with a highly variable linkage level. Turnover of links of specialists was driven by species turnover, whereas turnover of links among generalists took place through rewiring, i.e. by reshuffling existing interactions. However, in spite of this strong internal dynamics of species and links the network appeared overall stable. If this global stability-local instability phenomenon is general, it is a most astonishing feature of ecological networks.