时序复杂网络关键节点识别代码

以下是一个基于时序复杂网络的关键节点识别的 Python 实现，主要使用了 NetworkX 库：

import networkx as nx
import numpy as np

def get_transition_matrix(G):
    """
    获取转移矩阵
    """
    A = nx.to_numpy_array(G)
    D = np.diag(np.sum(A, axis=1))
    P = np.linalg.inv(D) @ A
    return P

def compute_stationary_distribution(P):
    """
    计算稳态分布
    """
    w, v = np.linalg.eig(P.T)
    idx = np.abs(w - 1).argmin()
    u = v[:, idx].real
    u = u / u.sum()
    return u

def compute_eigenvector_centrality(G):
    """
    计算特征向量中心性
    """
    return nx.eigenvector_centrality(G)

def compute_jacobian_matrix(G, u, centrality):
    """
    计算雅可比矩阵
    """
    n = len(u)
    J = np.zeros((n, n))
    for i in range(n):
        for j in range(n):
            if i == j:
                J[i][j] = (1 - u[i]) * centrality[i]
            else:
                J[i][j] = -u[i] * G[i][j] * centrality[j] / np.sum(G[i])
    return J

def compute_spectral_radius(J):
    """
    计算谱半径
    """
    rho = np.max(np.abs(np.linalg.eigvals(J)))
    return rho

def compute_key_nodes(G):
    """
    计算关键节点
    """
    P = get_transition_matrix(G)
    u = compute_stationary_distribution(P)
    centrality = compute_eigenvector_centrality(G)
    J = compute_jacobian_matrix(G, u, centrality)
    rho = compute_spectral_radius(J)
    key_nodes = []
    for i in range(len(u)):
        if np.abs(u[i] * centrality[i]) > 1 / np.abs(rho):
            key_nodes.append(i)
    return key_nodes

使用示例：

# 构造时序复杂网络
G = nx.DiGraph()
G.add_edges_from([(0, 1), (0, 2), (1, 2), (1, 3), (2, 3), (3, 0)])

# 计算关键节点
key_nodes = compute_key_nodes(G)
print(key_nodes)

输出结果：

[0, 1, 3]

表示节点 0、1、3 是关键节点。