the urn model [@godreche2002nonequilibrium; @0305-4470-38-19-R01] in which a site $p$ sends one of its particles to another site $q$ with rate $u_{pq}$ varying with both the departure and destination site \[Fig. \[fig:model\] (a)\].

Biquadratic transfer rate
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The transfer rate $u_{pq}$ can be determined to satisfy Eq. (\[eq:DApAp\]). A sequence of $\mathcal{N}$ particle jumps in or out of a site $p$ sums up to increase or decrease the number of particles at $p$ by $|\Delta m_p| \sim \sqrt{\mathcal{N}}$ as in random walks [@hughesbook]. Therefore the linear scaling in Eq. (\[eq:DApAp\]) can be reproduced if a site $p$ is selected $\mathcal{N} \propto m_p^2$ times per unit time. This reasoning leads us to the transfer rate $u_{pq}$ being [*biquadratic*]{} in the numbers of particles at $p$ and $q$. That is, $b=2$ in $$u_{pq}=u(m_p,m_q) = \dfrac{(m_p m_q)^b }{ \sum_{\ell} m_\ell^b}.
\label{eq:u}$$ If $b=0$, sites are uniformly selected, the number of particles at a site follows the Boltzmann distribution in the stationary state $p_\infty(m)=\rho^{-1} e^{-m/\rho}$ with the particle density $\rho\equiv M/N$ [@yakovenko2009colloquium]. If $b\neq 0$, sites are given unequal chances for particle hopping. If we consider particle hopping as occurring due to attraction between particles and assume that $u_{pq}$ is proportional to the sum 