  Dynamics among central sources (hubs) providing a resource and large number
of components enjoying and contributing to this resource describes many real
life situations. Modeling, controlling, and balancing this dynamics is a
general problem that arises in many scientific disciplines. We analyze a
stochastic dynamical system exhibiting this dynamics with a multiplicative
noise. We show that this model can be solved exactly by passing to variables
that describe the mass ratio between the components and the hub. We derive a
deterministic equation for the average mass ratio. This equation describes
logistic growth. We derive the full phase diagram of the model and identify
three regimes by calculating the sample and moment Lyapunov exponent of the
system. The first regime describes full balance between the non-hub components
and the hub, in the second regime the entire resource is concentrated mainly in
the hub, and in the third regime the resource is localized on a few non-hub
components and the hub. Surprisingly, in the limit of large number of
components the transition values do not depend on the amount of resource given
by the hub. This model has interesting application in the context of analysis
of porous media using Magnetic Resonance (MR) techniques.
