Complex network model of the phase transition on the wealth distributions - from Pareto to the society without middle class

A model of distribution of the wealth in a society based on the properties of complex networks has been proposed. The wealth is interpreted as a consequence of communication possibilities and proportional to the number of connections possessed by a person (as a vertex of the social network). Numerical simulation of wealth distribution shows a transition from the Pareto law to distribution with a gap demonstrating the absence of the middle class. Such a transition has been described as a second-order phase transition, the order parameter has been introduced and the value of the critical exponent has been found.

💡 Research Summary

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The paper proposes a novel way to model wealth distribution in a society by treating the social system as a complex network. In this representation each individual (or firm, etc.) is a vertex and the amount of wealth is assumed to be proportional to the vertex’s degree – the number of connections it has. The authors start from the classic Barabási‑Albert (BA) preferential‑attachment model, which naturally generates a scale‑free (Pareto) degree distribution, and then introduce a simple additional rule: a new edge can be attached to an existing vertex i only if the product r·k_i ≥ 1, where k_i is the current degree of i and r is a tunable parameter. In practice this means that only vertices whose degree exceeds a threshold 1/r are eligible to receive new links.

Simulations are performed with m=1 (each new vertex brings one edge), an initial seed of m₀=20 vertices, and a total of 20 000 vertices added. For each value of r the process is repeated 100–280 times to obtain reliable averages. Three regimes are identified:

1. r = 0 (no threshold) – the network grows exactly as the BA model, yielding a Pareto‑type degree distribution with exponent α≈0.63.
2. r < r_c (≈0.76) – the threshold is low enough that almost all vertices can still acquire new edges; the degree distribution remains Pareto‑like.
3. r > r_c – the threshold becomes restrictive. Low‑degree vertices are effectively excluded from further connections, leading to a pronounced gap η in the ranked degree distribution. In the reported example the gap spans degrees 4 to 65, meaning that no agents possess a “middle” amount of wealth. The network size shrinks (from ~20 000 to ~8 200 vertices) because many vertices become isolated and are not counted in the final degree ranking.

The authors interpret the emergence of the gap as a second‑order phase transition. They define the gap size η as an order parameter: η = 0 in the “ordered” (Pareto) phase and η > 0 in the “disordered” (gap) phase. By plotting η against the distance from the critical point (r – r_c) on double‑logarithmic axes they find a power‑law relationship η ∝ (r – r_c)^t with a critical exponent t ≈ 1.3. This exponent is obtained from a linear fit to the log‑log data and is comparable to exponents found in physical systems undergoing continuous transitions (e.g., magnetization near the Curie temperature).

The paper also explores the effect of edge weight distribution. When edge weights are drawn uniformly from {1,2,3} the results are essentially the same as when all edges have weight 1; only the numerical value of the critical threshold r_c shifts slightly, while the exponent t remains robust. This demonstrates that the phenomenon is driven primarily by the threshold rule rather than the specific weight distribution.

From a socio‑economic perspective, the parameter r can be viewed as a proxy for entry barriers in a market: capital requirements, regulatory hurdles, information asymmetry, or networking costs. A low r (low barriers) allows most agents to form connections, preserving a continuous wealth spectrum (including a middle class). A high r (high barriers) forces wealth accumulation to concentrate among already well‑connected agents, erasing the middle class and creating a polarized society. The model therefore suggests that policies aimed at reducing such barriers—improving access to credit, education, and networking opportunities—could keep the system below the critical threshold and maintain a more equitable wealth distribution.

In summary, the authors demonstrate that a minimal modification of the preferential‑attachment process—introducing a degree‑dependent connection threshold—produces a clear phase transition from a Pareto‑type wealth distribution to one with a macroscopic gap, which they interpret as the disappearance of the middle class. The identification of an order parameter and a critical exponent places the problem within the well‑established framework of statistical physics, offering a quantitative bridge between network theory and economic inequality research.