We consider a modification of the dynamical Erdős-Rényi random graph model, where connected components are removed (“frozen”) with a rate linearly proportional to their size. One may also view the time evolution of the list of component sizes of the graph as a multiplicative coalescent process with linear deletion (MCLD). This model exhibits self-organized criticality: the balance of creation and destruction keeps the model in a permanent state of criticality. We discuss this phenomenon using a surprising new “rigid representation” of the MCLD which also sheds some new light on Aldous’ representation of the scaling limit of the component sizes of the critical Erdős-Rényi graph as the collection of the excursion lengths of a reflected Brownian motion with parabolic drift. Joint work in progress with James Martin.