For our model's context, we theorize that this performance is modulated by two primary components:
Drawing from finance's geometric Brownian motion used for asset pricing, our equation becomes:
Where:
σ models performance volatility over time, expected to rise with node growth.
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Network performance is a multifaceted aspect of network operations, as highlighted in the endogenous model by Lin William Cong, (Cong, Li, and Wang 2021). In this model, the platform's productivity, represented as , evolves based on a geometric Brownian motion. Modeling is intended to simulate the system's performance trajectory, where epitomizes an asset or system's performance at a specified time.
A drift component, represented by , and
A random component, characterized by the product of the performance's standard deviation (σ) and a Wiener process .
Here, is a standard Brownian motion under physical measure, and both and are constants. is interpretation is broad. A positive shock to might encompass technological advancements, regulatory benefits, burgeoning user interest, or an expanding array of feasible platform activities. Naturally, this equation finds its solution as:
is the drift, accounting for performance's average growth rate.
is a random variable, representing real-world unpredictability.
Integrating these elements, the formula for encapsulates both deterministic and random system performance influencers, resulting in a temporally realistic behavior simulation.
In this endogenous performance model context, native cryptocurrency or tokens underscore system value, predominantly rooted in the platform's economic model. Their core value proposition interlinks with the user base's growth. This model's uniqueness resides in its assertion: both the token market price and user base size are exclusively driven by platform utility. This culminates in a stochastic equation for that necessitates modeling economic drift and volatility based on:
the sum of labor and financial contributions influenced by cyclical factors like financing seasons.
volatility, determined by node growth fluctuations and the Wiener process
Under such premises, network performance emerges as an escalating function. Ideally, this would dictate user and node growth, but in our model, these curves remain independent, mirroring expected market participant behavior. stands as a unique performance indicator, not directly affecting other models. In an ideal world, labor and investment would directly enhance network performance, but real-world dynamics introduce numerous factors, necessitating 's distinct parameter treatment.
Eliding detailed modeling from our model (Khvatov and Bogdanov 2023), we derived outcomes from relatively straightforward models for and . While this mirrors decentralized networks' nature aptly, it's pivotal to recall its inherent limitations owing to the system's core metrics' endogeneity. The network's growth is predominantly due to intricate external characteristics, shaped by high-tech investment seasonality, the economic milieu, and other factors.
