Equilibrium distributions in entropy driven balanced processes

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.