Bayesian NMF for Overlapping Community Detection

🔍 Overview

This algorithm performs overlapping community detection using Bayesian Non-negative Matrix Factorization (Bayesian NMF). It approximates a non-negative adjacency matrix V as a product of two lower-rank non-negative matrices W and H, and uses a Bayesian prior over the latent dimensions.

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⚙️ Class Definition

Class Name: BayesianNMF

This class implements a simple Bayesian NMF using multiplicative update rules and beta regularization.

class BayesianNMF:
    def __init__(self, n_clusters: int = 3, max_iter: int = 100, a: float = 1.0, b: float = 1.0, tol: float = 1e-5):
        ...

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📋 Parameters

Parameter	Type	Default	Description
n_clusters	int	3	Number of latent clusters (K)
max_iter	int	100	Maximum number of iterations
a	float	1.0	Gamma prior shape parameter
b	float	1.0	Gamma prior rate parameter
tol	float	1e-5	Convergence threshold for iterative updates

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💻 Using Example

from soft_clustering._bnmf._bnmf import BayesianNMF
import numpy as np

V = np.array([
    [0, 1, 0, 0, 0, 0],
    [1, 0, 1, 0, 0, 0],
    [0, 1, 0, 1, 0, 0],
    [0, 0, 1, 0, 1, 0],
    [0, 0, 0, 1, 0, 1],
    [0, 0, 0, 0, 1, 0]
], dtype=float)

model = BayesianNMF(n_clusters=2)
model.fit(V)
W = model.get_membership()

print(W)

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📥 Input / 📤 Output

• Input to fit(V):

  • V (np.ndarray): Non-negative adjacency matrix (N x N)

• Returns:

  • Internal W and H matrices

  • Soft membership matrix via get_membership()

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🛠️ Methods

• fit(V): Performs Bayesian NMF on matrix V

• get_membership(): Returns the matrix W of node memberships (N x K)

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📝 Implementation Notes

• Updates follow multiplicative rules

• Variational parameter β regularizes W and H

• Handles overlapping memberships naturally

• Fully unsupervised

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📚 Reference

This implementation is based on:
“Overlapping Community Detection using Bayesian Non-negative Matrix Factorization”
by D. Yang, J. Liu, and X. Tang.