<｜begin▁of▁sentence｜>### Instruction:
A data scientist wants to simulate a synthetic network where nodes belong to multiple communities with varying degrees of membership, to evaluate the performance of community detection algorithms. They decide to use the Mixed Membership Stochastic Block Model (MMSBM) to generate a graph with 6 nodes and 2 communities, where the probability of intra-community edges is high and the probability of inter-community edges is low. After simulating the graph, they aim to analyze it using network properties like PageRank. How can the MMSBM simulation be performed using graspologic, and how can the results be analyzed using NetworkX?
### Response:
import numpy as np
from graspologic.simulations import mmsbm
import networkx as nx

# Define the number of nodes and communities
n = 6
n_communities = 2

# Define the block connectivity matrix
p = [[0.8, 0.1], [0.1, 0.8]]

# Define the Dirichlet distribution parameter for mixed-membership
alpha = [0.2, 0.7]

# Use a random number generator for reproducibility
rng = np.random.default_rng(42)

# Generate the graph
A, labels = mmsbm(n, p, alpha, rng=rng, return_labels=True)

# Convert the adjacency matrix to a NetworkX graph
G = nx.from_numpy_array(A)

# Proceed with analysis on the NetworkX graph ...

# Example analysis: Compute the PageRank
pagerank_scores = nx.pagerank(G)

# Print results
print('PageRank scores:', pagerank_scores)
print('Membership labels matrix:\n', labels)
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