5.2.1 Node Ecosystem

Nodes are the linchpins of any decentralized platform. Acting as independent economic agents, they play a pivotal role in determining the dynamic behavior of tokens within the network. However, it's important to note that not all nodes are created equal, especially when it comes to token issuance. This differentiation led to the introduction of "Lucky Nodes," which hold a distinct position and responsibility within the network (Nsour and Sayama 2020).

Our tokenomics model is deeply rooted in the simulation of node growth, which provides insights into the projected evolution of nodes in the Decentralized Ecosystem (DEC) over specific time horizons. Nodes don't merely act as passive participants; they are entrusted with critical tasks such as processing transactions and upholding the network's operational sanctity. The allure of DEC tokens acts as a powerful incentive, motivating nodes to join and stay committed to the network (Tesfatsion, 2006).

In our endeavor to accurately simulate node growth, we rely on two foundational models:

• The Richards Model (Gürcan and Demirelli 2019): Renowned for its flexibility, the Richards model can adeptly represent various growth curve shapes. In the DEC context, we've incorporated a nuanced version of this model to encapsulate the complex dynamics of node growth. Traditionally, the Richards model operates on constant growth and attrition rates, culminating in a logistic growth curve. However, in our adaptation, we've added layers of depth by making both the growth rate r(t) and attrition rate a(t) time-dependent functions. This refinement not only makes the model more aligned with real-world dynamics but also enhances its predictive accuracy. The equations:

$$r(t) = \frac{r_0}{1 + k_r \cdot t} \ \ \ \ \ \ \ \tag{1}$$

$$a(t)=a_0⋅(1+k_a⋅t) \ \ \ \ \ \ \ \tag{2}$$

The general equation for node growth in this model is:

$$N_{nodes}(t) = \frac{N_{max}}{1 + (N_{max}/N_{0} - 1) \cdot exp(-\int (r(t) - a(t)) dt)} \ \ \ \ \ \ \ \tag{3}$$

• The Logistic Model (Merbis and Lodato 2022): Often termed the "S-shaped" model, this approach models node growth using a logistic curve, determined by specific differential equations. This curve's inherent shape and behavior allow us to capture the typical growth trajectory of nodes, especially in networks that experience saturation over time. The basic solution of the equation is given by the differential equation:

$$\frac{dN_{nodes}}{dt}=rN_{nodes}\ \left(1-\frac{N_{nodes}}K \right) \ \ \ \ \ \ \ \tag{4}$$

The resulting node growth N_nodes (t) is then described by the following formula:

$$N_{nodes}(t) = \frac{KN_0 \cdot e^{r(t+t_0)}}{K + N_0 \cdot (e^{r(t+t_0)} - 1)} \ \ \ \ \ \ \ \tag{5}$$

The figure below shows the output of node growth according to the Richards model:

Figure 72. Richards Model for Node Growth