In this paper we introduce a class of Markov models, termed best-effort networks, designed to capture performance indices such as mean transfer times in data networks with best-effort service. We introduce the so-called min bandwidth sharing policy as a conservative approximation to the classical max-min policy. We establish necessary and sufficient ergodicity conditions for best-effort networks under the min policy. We then resort to the mean field technique of statistical physics to analyze network performance deriving fixed point equations for the stationary distribution of large symmetrical best-effort networks. A specific instance of such net- works is the star-shaped network which constitutes a plausible model of a network with an overprovisioned backbone. Numerical and analytical study of the equations allows us to state a number of qualitative conclusions on the impact of traffic parameters (link loads) and topology parameters (route lengths) on mean document transfer time.