matrix.py

import numpy as np

# Step 1: Define the graph
edges = [
    (1, 3), (1, 4), (1, 5), (1, 6),
    (2, 3), (2, 4), (2, 5),
    (3, 1), (3, 4), (3, 6),
    (4, 5), (4, 6), (4, 7),
    (5, 7), (5, 8),
    (6, 5), (6, 7), (6, 9),
    (7, 8), (7, 9),
    (8, 9),
    (9, 8)
]

n = 9
A = np.zeros((n, n), dtype=float)
for u, v in edges:
    A[u - 1][v - 1] = 1

# Step 2: Power Iteration to compute Perron eigenvalue
def perron_lambda(A, tol=1e-9, max_iter=1000):
    n = A.shape[0]
    b = np.random.rand(n)
    b = b / np.linalg.norm(b)

    for _ in range(max_iter):
        b_new = A @ b
        b_new_norm = np.linalg.norm(b_new)
        if b_new_norm == 0:
            return 0
        b_new = b_new / b_new_norm
        if np.linalg.norm(b_new - b) < tol:
            break
        b = b_new

    lambda_perron = b.T @ A @ b
    return lambda_perron

# Step 3: Compute λ1
lambda1 = perron_lambda(A)
print(f"Perron eigenvalue λ1 ≈ {lambda1:.4f}")

# Step 4: Compute Influence
if lambda1 >= 1:
    omega = 1 / (1 - lambda1)  # May be negative if λ1 > 1
    r = np.sum(A, axis=1)
    I = (1 + omega) * r
    max_node = np.argmax(I) + 1
    print("Row Sum Vector r:", r)
    print(f"omega(λ1): {omega:.4f}")
    print("Influence Vector I:", I)
    print(f"Most Influential Node: Node {max_node} with value {I[max_node - 1]:.4f}")
else:
    print("λ1 < 1, influence vector may not be meaningful with this model.")