Model Overview

This model captures how fluctuations in the number of miners, M(t), and validators, V(t), influence the reward pool, R(t), and the overall answer quality, Q(t). The parameters are calibrated using three months of historical data from a similar public $TAO subnet. The resulting trajectories are illustrative of expected network dynamics.

Parameters & Variables:

• M(t): Number of active miners at time t

• V(t): Number of active validators at time t

• R(t): Reward pool distribution at time t

• Q(t): Quality metric of AI responses at time t

• λ: Rate of change (drift) in miner activity

• μ: Rate of change (drift) in validator activity

• σ: Standard deviation reflecting variability in reward distribution

Calibrated Values:

• Miner drift, λ = 0.002 day⁻¹ (growth rate of miners)

• Validator drift, μ = 0.001 day⁻¹ (growth rate of validators)

• Miner volatility, κₘ = 0.25 (reflects fluctuations in miner participation)

• Validator volatility, κᵥ = 0.15 (reflects fluctuations in validator participation)

• Reward pool standard deviation, σ_R = 10

• Miner/validator noise correlation, ρ ≈ 0.2 (captures the effect of regional or systemic events influencing both groups simultaneously)