Graph Value

Graph Value is a core metric in Local Protocol that quantifies the amount of value a user contributes to the network. It serves as a basis for allocating rewards to participants, ensuring that users are incentivized in proportion to their positive impact on the network’s connectivity and transaction flow.

Introduction to Graph Value#

In Local Protocol, Graph Value measures both economic activity and network influence for each participant. It is updated every time a transaction occurs, capturing changes in transaction volume and the participant’s centrality within the transaction graph.

The transaction graph is represented as a bipartite graph $G = (U, V, E)$, where:

• $U$: Set of producers,
• $V$: Set of buyers,
• $E$: Weighted edges representing transaction volumes.

Each transaction updates the metrics associated with a producer $u$ or buyer $v$, including:

• Total sum of edge weights ($W_u$ or $W_v$): Represents the cumulative transaction volume.
• Eigenvector Centrality ($x_u$ or $x_v$): Indicates a node’s influence within the graph.

Graph Value Update Rule#

For a given producer $u$, the Graph Value $G_u$ is calculated as:

$$G_u = G_u + \Delta G_u(\Delta W_u, \Delta x_u)$$

Where:

• $G_u$: Current Graph Value of producer $u$.
• $\Delta G_u(\Delta W_u, \Delta x_u)$: Change in Graph Value due to a transaction.
  • $\Delta W_u$: Change in transaction volume.
  • $\Delta x_u$: Change in eigenvector centrality.

Similarly, for a buyer $v$, the Graph Value $G_v$ is updated with changes in transaction volume and centrality:

$$G_v = G_v + \Delta G_v(\Delta W_v, \Delta x_v)$$

This update mechanism ensures that the Graph Value reflects both the transaction volume and the connectivity impact of each user.

Key Concepts#

1. Components of Graph Value#

• Transaction Volume ($W_u$ or $W_v$): Measures the total economic activity by summing the edge weights for a user.
  • The update $\Delta W_u$ or $\Delta W_v$ represents the volume of a completed transaction.
• Eigenvector Centrality ($x_u$ or $x_v$): Reflects a node’s importance in the network based on its connections to other influential nodes.
  • The update $\Delta x_u$ or $\Delta x_v$ accounts for changes in centrality due to new transactions.
• Reputation Score: A factor that can be included in Graph Value to account for a user’s historical performance and reliability. It serves as an additional measure of trust and value within the network.

Example of Graph Value Calculation#

Consider a producer $u$ with an initial Graph Value $G_u$, total transaction volume $W_u$, and eigenvector centrality $x_u$. After a transaction that increases $W_u$ by $\Delta W_u$ and $x_u$ by $\Delta x_u$, the new Graph Value is:

$$G_u = G_u + \alpha \cdot \Delta x_u + (1 - \alpha) \cdot \Delta W_u$$

• $\alpha$: A tunable parameter that controls the weight between centrality and transaction volume in Graph Value updates.

This formulation allows Local Protocol to adaptively adjust the weight of centrality and transaction volume based on network conditions and market maturity.

Next Steps#

The next topic will cover Sybil Resistance, showing how connectivity can provide a robust measure of realness for networks that do not have access to service-proofs or identity-proofs.