Sampling & Slashing

To ensure that the graph doesn't lose its security guarantees as new nodes enter the game, the network can randomly sample for service-proofs or service-approximations if proofs aren't available. If a node fails to provide their proofs, the network can slash the edge weights (tokens staked in the graph), and add an negative doping vector to the nodes that fail to provide proofs. This localized penalty system encourages self-policing and allows the network to remain secure without necessitating costly proofs for every transaction.

Key Concepts#

• Probabilistic Verification: Randomly samples transactions to maintain integrity.
• Penalization: Applies slashing penalties to users found guilty of dishonest actions.

Inverse Doping Vector#

When the network randomly samples a transaction and requests a service proof, the involved nodes must submit the required proof. If they fail to do so, we model this as an inverse doping vector in the eigenvector centrality (EC) calculation. Specifically, we decrease the EC scores of the nodes in question and remove the edge representing the fake transaction. This slashing not only impacts the penalized nodes but also affects their neighboring nodes, with the effect diminishing exponentially over longer paths in the graph.

$$\mathbf{x} = \frac{1}{\lambda_{\text{max}}} \mathbf{A} \mathbf{x} - \vec{d},$$

where $\vec{d}$ is a vector with positive entries corresponding to the penalized nodes, effectively reducing their EC scores.

For example, if node $u$ fails to submit a proof, the inverse doping vector $\vec{d}$ has a positive value $d_u$ at position $u$ and zeros elsewhere:

$$\vec{d} = (0, 0, \dotsc, d_u, \dotsc, 0)^\top.$$

The impact of this penalty propagates through the network due to the nature of the EC calculation and the edge weights $w_{uv}$ associated with the failed transaction are also decreased or set to zero:

$$w_{uv} \leftarrow w_{uv} \times (1 - \gamma),$$

Slashing Neighbors#

To further encourage self-policing, we can extend the penalty to nodes directly connected to the penalized node. This is modeled by adjusting the inverse doping vector to include these neighboring nodes with scaled penalties.

Let $N(u)$ denote the set of nodes directly connected to node $u$. We define the inverse doping vector $\vec{d}$ as:

$$d_i = \begin{cases} d_u & \text{if } i = u, \\ \alpha \times d_u & \text{if } i \in N(u), \\ 0 & \text{otherwise}, \end{cases}$$

where $0 < \alpha < 1$ is the decay factor representing the reduced penalty on neighboring nodes. For each node $i \in N(u)$, we can adjust the edge weights $w_{ui}$ associated with the neighboring node where $0 < \beta < \gamma$ is a smaller slashing factor for the connected edges.

$$w_{ui} \leftarrow w_{ui} \times (1 - \beta),$$

The effect of the penalty diminishes exponentially over longer paths in the network. Mathematically, this is inherent in the properties of the EC calculation. The further a node is from the penalized node, the less impact the inverse doping vector has on its EC score.

This decay can be adjusted through the choice of decay factor $\alpha$ and slashing factors $\gamma$ and $\beta$, allowing network designers to balance between strictness and leniency based on the desired security level.

This slashing mechanism encourages nodes to maintain genuine connections and discourages malicious behavior.

Next Steps#

The next topic, self-Policing, will explore how users can collectively regulate behavior within the network.