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The classical connectivity is typically used to capture the robustness of networks. Robustness, however, encompasses more than this simple definition of being connected. A spectral metric, referred to as the algebraic connectivity, plays a special role for the robustness since it measures the extent to which it is difficult to cut the network into independent components. We rely on the algebraic connectivity to study the robustness to random node and link failures in three important network models: the random graph of Erdos-Renyi, the small-world graph of Watts and Strogatz and the scale-free graph of Barabasi-Albert. We show that the robustness to random node and link failures significantly differs between the three models. This points to explicit influence of the network structure on the robustness. The homogeneous structure of the random graph of Erdos-Renyi implies an invariant robustness under random node failures. The heterogeneous structure of the small-world graph of Watts and Strogatz and scale-free graph of Barabasi-Albert, on the other hand, implies a non-trivial robustness to random node and link failures.