Title: Reward Function Design for Incentivizing Convergence in Decentralized Knowledge Graphs
Abstract
We propose a reward mechanism for decentralized systems built on the Collective Focus Theorem (CFT), where agents submit microblocks to a DAG representing partial computations toward a converging focus vector ( \pi ). The reward mechanism ensures that agents are fairly incentivized for submitting verified, convergence-aligned work. Our design combines multiple convergence indicators into a hybrid reward function, balancing simplicity, robustness, game resistance, and long-term epistemic value.
1. Introduction
The Collective Focus Theorem proves that token-weighted random walks in authenticated, directed graphs converge to a unique stationary distribution ( \pi ). To compute this distribution in a distributed fashion, agents (“neurons”) submit microblocks containing partial focus updates. However, to encourage meaningful participation and prevent manipulation, we need a verifiable and fair reward system.
This paper defines and compares several candidate reward functions and proposes a hybrid approach optimized for long-term convergence and resistance to manipulation.
2. Design Goals
The reward function must:
- Incentivize accurate computation toward ( \pi )
- Be locally verifiable
- Prevent reward extraction via oscillation or noise
- Reward both short-term and long-term contributions
- Scale with network size
3. Microblock Data Model
Each microblock includes:
- Parent block references
- Subgraph slice and cyberlinks
- Token state ( t_j )
- Input and output focus values: ( \pi^{(t)}, \pi^{(t+1)} )
- Proof hash and signature
4. Candidate Reward Functions
4.1 ( \Delta\pi ) Norm-Based Reward
[ R_1 = \alpha \cdot \sum_j |\pi_j^{(t+1)} - \pi_j^t| ]
Pros: Simple, easy to verify
Cons: Gameable by oscillation
4.2 Entropy Reduction Reward
[ R_2 = \beta \cdot (H(\pi^t) - H(\pi^{t+1})) ]
Where entropy is ( H(\pi) = -\sum_j \pi_j \log \pi_j )
Pros: Rewards semantic sharpening
Cons: Computationally heavier
4.3 Cosine Similarity to Target
[ R_3 = \gamma \cdot \text{cos}(\pi^{(t+1)}, \pi^*) ]
Pros: Alignment with oracle( \pi^* )
Cons: Requires trusted reference; hard for local compute
4.4 Spectral Gap Improvement
[ R_4 = \delta \cdot (\lambda_2^t - \lambda_2^{t+1}) ]
Where ( \lambda_2 ) is the second eigenvalue of the transition matrix
Pros: Measures global convergence speedup
Cons: Expensive and non-local
4.5 Predictive Alignment Reward
[ R_5 = \epsilon \cdot \text{align}(\pi_j^{(t+1)}, \pi_j^{T}) ]
Pros: Favors early correct contributions
Cons: Requires delayed validation
5. Composite Hybrid Model
We propose the following hybrid reward function:
[ \text{Reward} = \alpha \cdot \Delta\pi + \beta \cdot \Delta H + \gamma \cdot \text{DAGWeight} + \epsilon \cdot \text{AlignmentBonus} ]
Where:
- ( \Delta\pi ): total L1 delta in focus vector
- ( \Delta H ): entropy drop
- DAGWeight: number of descendant blocks referencing this one
- AlignmentBonus: comparison with future ( \pi^T ) (optional, delayed)
6. Implementation Strategy
- Fast local rewards use ( \Delta\pi ) and ( \Delta H )
- Checkpoints add alignment and spectral verification bonuses
- Validators sample and verify blocks probabilistically
7. Conclusion
This hybrid function aligns incentives with convergence, penalizes noise, rewards foresight, and supports scaling. It anchors computation in provable improvements to ( \pi ), the collective intelligence substrate.
Future work includes benchmarking reward trajectories and testing robustness against coordinated spam and adversarial updates.