5.6.1 Node-Base Demand

In the realm of blockchain, nodes play a pivotal role, not only validating transactions but also upholding the sanctity and integrity of the entire system. Their function is paramount, and in return for this service, they incur operational expenses. These costs, for many networks, are offset by token rewards, which both serve as an incentive and a means to balance the financial outlay associated with running a node. Thus, understanding the token demand from the perspective of these crucial entities is vital. It brings forth clarity on how the intrinsic dynamics of token demand and supply play out in real-time within the network.

Modeling the Demand Dynamics:

• Basics of Node Operation and Token Pricing: The genesis of our model is grounded in the intrinsic value of the token, which is mapped to its demand by the nodes. However, to initiate our calculations, we first need a grasp on the token's fiat currency price. While a profound analysis would embed myriad variables, for brevity, we've hinged our model on the Sigmoid Model of the Token Price. This choice of model is not arbitrary. The sigmoid function, with its characteristic S-shape, encapsulates the trajectory of many cryptocurrencies: a slow outset, followed by an acceleration, culminating in a gradual tapering off.

• Token Price Modeling with Sigmoid Function: Mathematically, the token price over time is represented as:

$$P_t^{sigmoid}= \frac{P_t^{max}}{1+e^{-k_S\cdot t-t^*}}+P_t^0 \ \ \ \ \ \ \ \ \tag{31}$$

Here:

o controls the curve's steepness.

o marks the midpoint of the time period.

o and are the peak and initial token values respectively.

Visualize this curve and one observes the lifecycle of many well-known cryptocurrencies: an initial period of obscurity, a rush of adoption, followed by a phase of maturity.

Figure 84 Token Price over time

• Node Token Demand Dynamics: The crux of the token demand stems from the nodes' operational costs. This cost is a function of myriad variables, like the direct expenses of running a node and the associated token requirements, both of which evolve over time. Thus, our model captures this dynamic demand through the equation:

$$D_{node}^{sigmoid} = \frac{D_{node}^0 \cdot (1+G_r)^{t^*}}{P_t^{sigmoid} + P_t^{extra}} \ \ \ \ \ \ \ \ \tag{32}$$

Where:

o is the node's cost growth rate.

o symbolizes any additional node costs, possibly due to factors like market competition or external regulatory constraints.

• Incorporating Network Effects: While the above model captures the fundamental node operations, the interplay of network effects, which escalate as the number of nodes increase, cannot be sidelined. To embed this crucial aspect, the model is refined to:

In this equation, is the influence of the network effect. The equation elegantly captures how, with time, the token requirements for operational expenses stabilize as the token appreciates.

Figure 85 Token Demand by Nodes over Time