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Wealth inequality, though age-old, has seen a substantial increase since the early 21st century [1]. The distributions of wealth and income across countries present a universal pattern, often manifesting as a two-class division [2]. This suggests that fundamental mechanisms underpin the emergence of economic inequalities. Agent-based models, which allow us to define the rules of interaction between economic agents, are well-suited for studying economic systems and analyzing their emergent properties. These models are typically applied in a mean-field manner, where agents are randomly selected for pairwise interactions, and are often restricted to a conservative market. Incorporating economic growth into such models is a natural extension. A common approach to modeling economic growth is to assume each agent’s wealth grows through a stochastic process. Another interesting aspect is the topological organization of agents, where the structure of interactions can evolve dynamically. In this work, we investigate a recently proposed dynamic complex network agent-based model [3] within a growing economic scenario. The model evolves through three alternating processes: independent stochastic wealth growth  of each agent, wealth exchanges between connected agents, and rewiring of connections within the complex network. Wealth growth for each agent follows a stochastic process with two parameters: a drift term $\mu$, representing economic growth, and volatility $\sigma$, reflecting heterogeneity in productivity. For wealth exchange, we employ the Yard-sale model, where the wealth exchanged between agents is given by $\Delta \omega(t) = min[\alpha_i\omega_i(t), \alpha_j\omega_j(t)]$, where $\omega_i(t)$ represents agent i's wealth and $\alpha_i$ its risk factor. Thus, the wealth of an agent after one time step is $\omega_i(t+1) = \omega_i(t)(\mu + \sigma dW) + \Delta \omega_i(t)$, where $dW$ is a Wiener process, and $\Delta \omega_i(t)$ is the net wealth traded by agent i in that step. We analyze the results for different values of a social protection factor $f$, which favors the poorer agent in each transaction. For $f = 0$, our results show condensation of wealth and connections in a single agent, independent of $\mu$ and $\sigma$. Interestingly, even a small value of social protection ($f = 0.01$) favors agents from the middle and upper classes, leading to the formation of hubs in the network [3]. However, for sufficiently large values of $\sigma$, wealth condensation reappears. Larger values of $f$, on the other hand, makes the economic growth becomes more significant: while increasing $\mu$ reduces inequality, increasing $\sigma$ has the opposite effect. In this context, economic growth benefits the poorest agents only when strong social protection is in place.


Acknowledgments
This work was supported by the Brazilian funding agencies CNPq.
References
[1] L. Chancel, T. Piketty, Global income inequality, 1820–2020: the persistence and mutation of extreme inequality, Journal of the European Economic Association 19 (6) (2021) 3025–3062.
[2] B. K. Chakrabarti, A. Chakraborti, S. R. Chakravarty, A. Chatterjee, Econophysics of income and wealth distributions (2013).
[3] Kohlrausch, Gustavo L., and Sebastián Gonçalves. "Wealth distribution on a dynamic complex network." Physica A: Statistical Mechanics and its Applications (2024): 130067.