Weighted trade network in a model of preferential bipartite transactions

Using a model of wealth distribution where traders are characterized by quenched random saving propensities and trade among themselves by bipartite transactions, we mimic the enhanced rates of trading of the rich by introducing the preferential selection rule using a pair of continuously tunable parameters. The bipartite trading defines a growing trade network of traders linked by their mutual trade relationships. With the preferential selection rule this network appears to be highly heterogeneous characterized by the scale-free nodal degree and the link weight distributions and presents signatures of nontrivial strength-degree correlations. With detailed numerical simulations and using finite-size scaling analysis we present evidence that the associated critical exponents are continuous functions of the tuning parameters. However the wealth distribution has been observed to follow the well-known Pareto law robustly for all positive values of the tuning parameters.

Figure 1

(Color online) (a) Plots of $⟨x\left(\lambda \right)⟩\left(1-\lambda \right)$ vs $\lambda$ for $\alpha =\beta =0$ (black), 1/2 (red), 1 (green), 3/2 (blue), and 2 (magenta) for $N=1024$ ($\alpha$ increases from top to bottom). (b) The product function $⟨x\left(\lambda \right)⟩\left(1-\lambda \right){\lambda }^{-\chi \left(\alpha \right)}$ is plotted with $\lambda$ using $\chi \left(\alpha \right)=0.15$, 0.35, 0.57, and 0.80 for $\alpha =1/2$, 1, 3/2, and 2, respectively, using the same colors as in (a).

Figure 2

(Color online) (a) Wealth distribution $P\left(x,N\right)$ vs $x$ for $\alpha =\beta =0$, 1/2, 1, and 2 and for $N=256$ (black), 1024 (red), and 4096 (blue) ($N$ increases from left to right). The slopes of these curves yield $\nu =1.00\left(3\right)$ consistent with the Pareto law. (b) $P\left(x,N\right)$ for $\left(\alpha ,\beta \right)=\left(\infty ,0\right)$ (black) and $\left(0,\infty \right)$ (red) for $N=1024$ which almost overlapped. (c) $P\left(x,N\right)$ for $\left(\alpha ,\beta \right)=\left(\infty ,\infty \right)$, the distribution is uniform followed by a hump due to transactions between the richest and the next richest traders only.

Figure 3

The phase diagram in the positive quadrant of the $\left(\alpha ,\beta \right)$ plane. The Pareto law is valid in the entire region. The origin corresponds to the CCM model where the trade network is a random graph. At the corners $\left(\infty ,0\right)$ and $\left(0,\infty \right)$ the richest trader trade in every transaction, so that the network is a starlike graph but the wealth distribution still follows Pareto law as shown in Fig. 2b. However at the corner $\left(\infty ,\infty \right)$ the trade takes place between the top two richest traders and the network shrinks to a single dimer. The wealth distribution here is uniform followed by a hump as shown in Fig. 2c.

Figure 4

(Color online) Growth of the trade network. (a) Plot of the average size of the giant component $⟨s_m\left(\rho ,N\right)⟩$ with the link density $\rho$, for $\alpha =\beta =1$ and for $N=128$ (black), 256 (red), 512 (blue), and 1024 (pink), ($N$ increases from right to left). The inset shows a data collapse of the same plots with $\rho N^{\theta }$, and $\theta =0.88$. (b) The percolation link density ${\rho }_c\left(N\right)$ is plotted with $N^{-\theta \left(\alpha \right)}$ where $\theta \left(\alpha \right)=0.88$, 0.92 and 1 for $\alpha =\beta =1$ (black), 3/4 (red), and 1/2 (blue) respectively ($\alpha$ increases from left to right). The inset shows plot of $\theta \left(\alpha \right)$ with $\alpha$.

Figure 5

(Color online) (a) The finite-size scaling of the degree distributions $P\left(k,N\right)$ with $\alpha =\beta =1$ for $N=256$ (black), 1024 (red), and 4096 (blue) and for $⟨k⟩=1$ ($N$ decreases from left to right). Direct measurement of slopes give ${\gamma }_k\left(1\right)=2.18\left(3\right)$. The best data collapse corresponds to ${\eta }_k\left(1\right)=1.62$ and ${\zeta }_k\left(1\right)=0.75$ giving ${\gamma }_k\left(1\right)=2.16\left(3\right)$. (b) The plot of ${\gamma }_k\left(\alpha \right)$ vs $\alpha$.

Figure 6

(Color online) Probability distribution $P\left(w,\alpha \right)$ of the link weights. (a) Plots for the $N$-clique graphs with $N=64$ and for $\alpha =0$ (black), 1/4 (green), 1/2 (blue), 3/4 (magenta), and 1 (red) and $\beta =1$ always (the maximum value decreases with increasing $\alpha$). (b) Plots for $⟨k⟩=5$ with $\alpha =\beta =1$ and for the system sizes 128 (black), 256 (red), and 512 (blue) ($N$ decreases from left to right). Direct measurement of slopes gives 2.52, 2.53, and 2.51, respectively.

Figure 7

(Color online) (a) Nodal strength distributions $P\left(s,N\right)$ with strength $s$ with $\alpha =\beta =1$, for $N=256$ (black), 1024 (red), and 4096 (blue) ($N$ increases from left to right) and for $⟨k⟩=5$. Direct measurement of the slopes of these curves gives ${\gamma }_s=2.50\left(5\right)$. (b) The Finite-size scaling analysis of the data (using same colors) in (a) gives ${\eta }_s=1.64$ and ${\zeta }_s=0.67$ estimating the value of ${\gamma }_s={\eta }_s/{\zeta }_s=2.45\left(5\right)$.

Figure 8

(Color online) (a) Plot of nodal strength $⟨s\left(k\right)⟩$ with degree $k$ for $N=16384$ and for $\alpha =\beta =1/2$ (black), 3/4 (green), 1 (blue), 5/4 (magenta), and 3/2 (red) ($\alpha$ decreases from left to right). The slopes $\varphi \left(\alpha \right)$ of these curves are plotted in the inset. (b) Plot of average wealth $⟨x\left(k\right)⟩$ with degree $k$ for $N=16384$ and for $\alpha =\beta =1/2$ (black), 3/4 (green), 1 (blue), 5/4 (magenta), and 3/2 (red) ($\alpha$ increases from left to right). The slopes $\mu \left(\alpha \right)$ of these curves are plotted in the inset.