Abstract: Networks are a general framework by which we can describe a large class of many-body systems of interacting elements. We will propose a model that describes in a stylized way how local breakdowns due for example to congestion problems may shape the structure of a network. The model converges to a self-organized critical stationary state in which the network structure results as a consequence of avalanches of rewiring processes. Depending on the model's specification, we obtain either single scale or scale-free networks. We characterize in detail the relation between the statistical properties of the network and the nature of the critical state, by computing the critical exponents. The model also displays a non-trivial, sudden, collapse to a complete network.