Anomaly, Class Division, and Decoupling in Income Dynamics

TL;DR

The paper presents a minimal income-dynamics framework based on a heterogeneous Bouchaud–Mézard (HBM) model, where regional growth-rate heterogeneity is encoded by a binary mixture and controlled by growth-rate assortativity 𝒜 and concentration 𝓡. By analytic treatment on a 1D ring and numerical exploration on Watts–Strogatz networks, it derives closed-form approximations for the Hellinger distance and Gini index, and reveals how strong regional growth-rate segregation yields bimodality and persistent spatial correlations. Small-world shortcuts disrupt segregation and bimodality, providing a mechanism by which network structure can mitigate inequality patterns akin to those observed in historical global income distributions. The results connect to broader narratives of global inequality, linking network topology to phase-separation-like dynamics and offering a quantitative lens for understanding eras of rising, then stabilizing, segregation in income distributions. The framework points toward network-aware strategies to alleviate income inequality by altering connectivity patterns that sustain regional disparities.

Abstract

Economic inequality emerges from the interplay between regional growth-rate differences and the interaction network that couples regions. We propose a minimal income-dynamics model, where heterogeneity is governed by growth-rate assortativity $\mathcal{A}$ and regional concentration $\mathcal{R}$, allowing us to quantify the spatiotemporal patterns of empirically observed log-income distributions. To systematically analyze these patterns, we derive closed-form approximations for the Hellinger distance and the Gini index in limiting configurations. Our findings highlight the spatial segregation of growth rates as a key driver of economic class division and demonstrate how small-world shortcuts in the underlying network can disrupt this segregation. Finally, our framework provides a robust explanation for the bimodality and strong regional correlations found in global income distributions.

Figures (18)

• Figure 1: (a) Schematic illustration of income dynamics with a binary mixture of regional growth rate in the HBM model: ${\color{red}\bullet}~(\alpha_{+}, \text{red})$ and ${\color{blue}\bullet}~(\alpha_{-}, \text{blue})$ for $\alpha_\pm=\alpha\pm \Delta \alpha$ and income ($C$) transfer (either $\rightarrow$ or $\leftarrow$) between two nearest-neighboring sites. (b)-(c) Snapshots of spatiotemporal patterns for a top-rich/bottom-poor 10% (orange/green) class are taken from a single run for two extreme configurations: ($\mathcal{A}_{\min},\mathcal{A}_{\max}$): The position index $n$ is shown in horizontal from left to right, and time $t$ is in vertical from top to bottom. Here $N=10^3,\alpha=10^{-2},\Delta\alpha=10^{-3},\beta^2=10^{-3},~\hbox{and}~ J=10^{-1}$ in Eq. \ref{['eq-HBM']}. (d) Random pair swapping trajectories of $(\mathcal{A}, \mathcal{R})$. A path starts from two extreme configurations with $(\mathcal{A}_{\rm min/max},\mathcal{R}_{\rm min/max})$ (see insets). Each random pair-swapping trial is represented by color gradation. The interval of simulation samples ($\circ$) is 0.1 in [$\mathcal{A}_{\rm min},\mathcal{A}_{\rm max}$].
• Figure 2: Configuration effect on decoupling by Hellinger distance $h$: (a) $h$ in Eq. \ref{['eq-h']} plotted at $t=10^5$ against $\mathcal{A}$ with $\mathcal{R}(\mathcal{A}$), see Fig. \ref{['fig1-model']}(d). Selected snapshots for unimodal and bimodal distributions: (b), (c), (d), and (e) are the cases of $(\mathcal{A}, \mathcal{R})=(-1.0, 0.0),~(-0.5,0.0),~(0.5,0.7),~\hbox{and}~(1.0,1.0)$ for $t=10^4$, respectively. Here, $N=10^4$, results are obtained by 128 runs, and the other parameters are the same as Fig. \ref{['fig1-model']}.
• Figure 3: Configuration effect on inequality by Gini index $g$ against $t$: Each color reflects the selected samples of $\mathcal{A}$ with $\mathcal{R}$ in Fig. \ref{['fig1-model']} (d). Path 1 (a) of $(\mathcal{A}_{\min}, \mathcal{R}_{\min})\to (0,0)$ and Path 2 (b) of $(\mathcal{A}_{\max}, \mathcal{R}_{\max})\to (0,0)$, respectively. As $t$ elapses, the system exhibits normal diffusion (below $t_1$), sub-diffusion ($t_1\le t\le t_2$) and finally reaches the configuration-effect dominant diffusion (above $t_2$). Insets in (a) and (b) are ${\rm Var}(X)$ and ${\rm Var}(X_{\rm rich})$, respectively, where the guided lines are drawn for normal diffusion and sub-diffusion as described, respectively. Lorenz curves (c) and (d) of $S$ against $f$ at $t=10^4$ correspond to Fig. \ref{['fig2-h']}(b)-(d) and (c)-(e), where plain and hatched shadow regions illustrate the contributions of class division and diffusion to $g$, respectively.
• Figure 4: SW effect on the HBM model: Spatiotemporal patterns of top-rich (orange)/bottom-poor (green) 10% classes over positional index $n$ for a fully separated configuration with rewiring probability (a) $p= 0.1$ and (b) $p=0.5$ for adding shortcuts. (c) Snapshots of log-income distributions for various $p$ values at $t=10^4$. The inset displays income-level segregation $\Delta\mu$ against $p$. Here we employ the WS network with $k=4$, and $N=10^3,\Delta\alpha=10^{-3}$ in (a) and (b); $N=10^4,\Delta\alpha=5\times10^{-3}$ in (c). The other parameters are the same as Fig. \ref{['fig1-model']} (b) and (c).
• Figure B1: Visualization of Lorenz curve and decomposition for the case of a dual log-normal mixture with large decoupling: Entire Lorenz curve $\mathcal{L}$ (left); Decomposition of $\mathcal{L}$ (right), where blue and red solid lines show rescaled Lorenz curves $\mathcal{L}_1$ and $\mathcal{L}_2$, and $S_1$ and $S_2$ represent total income share from each log-normal distribution. Light-blue and light-orange shadowed areas show contributions of between- and within-inequality on $g$ for $f_1=f_2=1/2$.