Multiplicative coalescent with linear deletion: a rigid representation
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
This model exhibits self-organized criticality: the balance of
creation and destruction keeps the model in a permanent state of
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.