the birth and death of product categories, leading to expansion or contraction of the product space. Also intrinsic fitness of products can be considered. Another natural development would be to assume that capital transfers occur on a network of products linked through heterogeneously weighted development pathways [@hidalgo2007product; @hidalgo2009building; @hausmann2011network].

This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korean Government (No. 2013R1A1A201068845 and No. 2016R1A2B4013204).

Derivation of the single-site distribution in the stationary state of the urn model with $b=2$ {#sec:pm}
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In Eq. (\[eq:factor\]), the factorized form of the configuration probability $p_\infty(\{m\}) = Z_{N,M}^{-1} \prod_i f_2 (m_i)$ with $f_2(m) = m^{-2}/\zeta(2)$ leads us to find that the grand partition function is given by ${\mathcal{Z}}_N (z) = \sum_{M=0}^\infty z^M Z_{N,M}= F_2(z)^N$ with $F_2(z)$ the generating function of $f_2(m)$ represented as $F_2(z)\equiv\sum_{m=1}^\infty  z^m f_2(m) = \sum_{m=1}^\infty {z^m \over \zeta(2) m^2} = {{\rm Li}_2 (z) \over \zeta(2)}$, where ${\rm Li}_b(z)$ is the poly-logarithm function ${\rm Li}_b(z) = \sum_{m=1}^\infty z^m m^{-b}$. This series converges for $|z|\leq 1$ and expanded around $z=1$ as [@PhysRev.83.678] $$F(z=e^{-\alpha}) = 1 + {\alpha \log \alpha -\alpha  \over \zeta(2)} + \sum_{n=2}^\infty {(-1)^n \over n!} {\zeta(2-n) \over \zeta(2)} \alpha^n.
\label{eq:Fz}$$

The partition function 