  For entropy driven balanced processes we obtain final states with Poisson,
Bernoulli, negative binomial and P\'olya distributions. We apply this both for
complex networks and particle production. For random networks we follow the
evolution of the degree distribution, $P_n$, in a system where a node can
activate $k$ fixed connections from $K$ possible partnerships among all nodes.
The total number of connections, $N$, is also fixed. For particle physics
problems $P_n$ is the probability of having $n$ particles (or other quanta)
distributed among $k$ states (phase space cells) while altogether a fixed
number of $N$ particles reside on $K$ states.
