5.2.2 User Growth and Retention Modeling

For decentralized platforms, understanding user behavior patterns is paramount. The growth, activity, and retention of users are the lifeblood of such platforms, significantly influencing their trajectory and potential success.

The LGND (Logistic-Gompertz-Node Driven) Model:

At the heart of our user growth modeling is the LGND model. This model is particularly favored for its adaptability to typical user growth patterns observed in network systems (Estrada and Bartesaghi 2022). The LGND model beautifully amalgamates three pivotal aspects:

• Logistic and Gompertz Growth Curves: These are classical representations of growth, where the logistic curve models resource-driven growth in a limited environment, and the Gompertz curve captures growth that's slowest at the outset and culmination. Both curves provide foundational insights into how users might join and interact with a platform. The equation for logistic growth is:

$$U_{logistic}(t) = \frac {U_0}{\widehat{U}} \cdot \frac{e^{r \cdot (t - t_0)}} {1 + U_0 \cdot e^{r \cdot (t - t_0) - 1}} \ \ \ \ \ \ \ \tag{6}$$

Here,

The equation for Gompertz growth is:

$$U_{gompertz}(t) = U_0 + (\widehat{U} - U_0) * e^{\epsilon \cdot e^{r * (t - t0)}} \ \ \ \ \ \ \ \tag{7}$$

• Node-Driven Growth: This represents the network effect, suggesting that as the network (or number of nodes) grows, it becomes increasingly valuable, thus attracting even more users. This effect captures the synergistic growth observed in many successful platforms, where user acquisition begets more user acquisition. The equation for user-driven node growth is:

$$U_{Nodes}(t) = U_0 + \frac{N_{Nodes}}{TotalNodes} \cdot (\widehat{U} - U_0) \ \ \ \ \ \ \ \tag{8}$$

• Realistic Behavior Amendments: Acknowledging the unpredictable nature of real-world scenarios, this model also factors in both negative events, like regulatory changes or security breaches, and positive stimuli, like marketing campaigns. These amendments ensure the model isn't just theoretically sound but also practically relevant.

However, the LGND model's precision hinges on the accurate selection of its coefficients and parameters. This choice can be daunting, given the intricate and evolving nature of decentralized systems.

Figure 73. User Growth Model (LGND + Realistic Behavior)

The Alternative User Engaging Model:

This model takes a more explicit approach to user growth, grounding its predictions in the platform's carrying capacity and the impact of marketing initiatives. By tying user growth directly to external factors and actionable strategies, this model offers a tangible way to evaluate and optimize user acquisition efforts. But its strength can also be a limitation, as it might lack flexibility in the face of unexpected system dynamics or data unavailability. The actual calculations hinge heavily on the estimation of the platform's performance (A_t), which can either be driven by the endogenous factors or be a consequence of the exogenous influences channeled through node growth and user engagement:

• Calculation of Carrying Capacity K (K. Wang et al. 2022): Originating from population biology, the concept of carrying capacity embodies the maximum population size that a specific environment can sustain over time. For our user growth model within a network, the carrying capacity is considered a theoretical ceiling or a normalizing component that shapes the user growth equation:

$$K(t) = f (A_t, \widehat{U}, N_{nodes}) + \psi(t) \ \ \ \ \ \ \ \tag{9}$$

• Calculation of Marketing Impact Φ: This facet of the algorithm simulates how marketing activities contribute to user base growth. The Marketing Impact (Φ) is treated as a function comprising three variables: the platform's performance (At), the number of nodes (Nnodes), and the impact of marketing activities, which is portrayed by an exponential decay function. The implementation steps of the algorithm essentially rely on these components:

$$Ф(t) = \sum_{i=0}^{n} C_i \cdot \frac {exp(-(t - T_i)^2} {2 \cdot σ_i^2} \ \ \ \ \ \ \ \tag{10}$$

• User Base Calculation: The model leverages a difference equation to portray the growth of the user base:

$$\frac{dU}{dt} = r_{users} \cdot U \cdot (1 - \frac{U}{K_{users}}) + \Phi \ \ \ \ \ \ \ \tag{11}$$

where:

Implications and Practical Application:

User growth isn't merely a metric; it's a reflection of the platform's vitality. Active user engagement drives platform success, fostering a vibrant community, increasing transaction volumes, and bolstering network security. Through the LGND model, we gain not just a predictive tool but also a strategic compass. By simulating user growth and understanding its underlying drivers, platforms can tailor their strategies, optimize their resources, and anticipate challenges. This proactive approach ensures that the platform remains resilient, adaptive, and poised for sustainable growth.