Basic Example Graph

You can visualize a network as a bipartite graph $G = (U, V, E)$ representing producers $U$ and buyers $V$ as nodes, with weighted edges $E$ capturing transactions between them. Edge weights $w(u, v)$ track the cumulative fees contributed by producer $u$ from transactions with buyer $v$.

To capture both the economic activity and the network influence of each node, we introduce a graph value metric that incorporates eigenvector centrality. For each producer $u$, we define the graph value $G_u$ as:

$$G_u = \alpha \cdot |EC_u| + (1 - \alpha) \cdot \sum_{(u,v)\in E} w(u,v)$$

Where:

• $|EC_u|$ is the absolute value of the eigenvector centrality of node $u$
• $\alpha$ is a tunable parameter between 0 and 1
• $\sum_{(u,v)\in E} w(u,v)$ is the sum of all edge weights connected to $u$

Using the absolute value in the calculation:

Ensures Non-negativity: This approach guarantees that all centrality measures contribute positively to the graph value, aligning with the typical requirements of token distribution mechanisms where negative contributions are not practical.

Preserves the Interpretative Integrity: While taking the absolute value can sometimes mask the directionality (if relevant), in many network applications, especially those involving token distributions or influence measures, the magnitude of centrality is more critical than its direction.

The eigenvector centrality $EC_u$ is calculated by solving the eigenvector equation:

$$\lambda \mathbf{x} = A\mathbf{x}$$

Where:

• $A$ is the adjacency matrix of the graph
• $\lambda$ is the largest eigenvalue
• $\mathbf{x}$ is the corresponding eigenvector

The eigenvector centrality of a node is its corresponding value in the normalized eigenvector $\mathbf{x}$.

This graph value metric $G_u$ combines the node's economic activity (represented by the sum of edge weights) with its structural importance in the network (represented by its eigenvector centrality). The parameter $\alpha$ allows for tuning the balance between these two factors.

---

Simple Example of a Dynamic Graph

Let's simulate a small network with 2 producers (P1, P2) and 3 buyers (B1, B2, B3). We'll use $\alpha = 0.5$ for our calculations.

In Python, we can define a function calculate_graph_values that performs our calculations with the NetworkX package.

Code
1import numpy as np2import networkx as nx3
4def calculate_graph_values(G, alpha=0.5):5    # Ensure consistent node ordering6    node_order = ['P1', 'P2', 'B1', 'B2', 'B3']7    8    # Create adjacency matrix9    A = nx.to_numpy_array(G, nodelist=node_order)10    11    # Calculate eigenvector centrality12    eigenvalues, eigenvectors = np.linalg.eig(A)13    largest_index = np.argmax(eigenvalues)14    eigenvector = eigenvectors[:, largest_index].real15    16    # Normalize eigenvector17    eigenvector = np.abs(eigenvector) / np.linalg.norm(eigenvector)18    19    # Calculate graph values for producers20    graph_values = {}21    for i, node in enumerate(node_order):22        if node.startswith('P'):23            economic_activity = sum(G[node][neighbor]['weight'] for neighbor in G[node])24            graph_values[node] = alpha * eigenvector[i] + (1 - alpha) * economic_activity25    26    return A, eigenvector, graph_values

Initial transaction graph:#

We will define an initial graph:

Code
1# Create initial graph2G = nx.Graph()3G.add_edge('P1', 'B1', weight=2)4G.add_edge('P2', 'B2', weight=3)5G.add_edge('P2', 'B3', weight=1)

When we run the graph through our calculate_graph_values function as:

Code
A, eigenvector, graph_values = calculate_graph_values(G)

we produce the following results.

P1	P2	B1	B2	B3
P1	0	0	2	0	0
P2	0	0	0	3	1
B1	2	0	0	0	0
B2	0	3	0	0	0
B3	0	1	0	0	0

Solving $\lambda \mathbf{x} = A\mathbf{x}$, we get the normalized absolute eigenvector value for each node:

Node	Value
P1	0.0000
P2	0.7071
B1	0.0000
B2	0.6708
B3	0.2236