Implementation of Newman Algorithm for Zachary Network Division in Complex Networks

Newman's algorithm is a classic community detection algorithm widely used for analyzing community structures in complex networks. This algorithm iteratively optimizes the modularity metric to partition network nodes, resulting in densely connected nodes within the same community and sparse connections between different communities. The implementation typically involves matrix operations and greedy optimization techniques to maximize modularity gain.

Zachary's karate club network is a well-known sociological dataset containing 34 nodes and 78 edges, representing social relationships among club members. This network serves as a standard benchmark for testing community detection algorithms, with its ground truth division reflecting the actual split of the club into two factions.

The algorithm implementation primarily consists of two files: Newman_Zachary: The core algorithm file containing key functions for modularity calculation, matrix operations, and community merging. The implementation follows a greedy strategy where community pairs yielding maximum modularity increase are progressively merged until no further optimization is possible. Key functions include modularity_matrix() for matrix computation and merge_communities() for optimal pair selection. Zachary-E: The experimental data file storing Zachary network data in edge-list format. Each line contains a pair of node identifiers representing connection relationships, typically loaded using read_edge_list() function for network initialization.

The implementation workflow consists of three main phases: Initialization: Treat each node as an independent community and compute initial modularity using adjacency matrix and degree vectors. Iterative Merging: Evaluate all possible community pairs through nested loops, calculate modularity gains using delta_Q formulas, and execute the optimal merge operation. Termination: Stop iterations when modularity cannot be further improved, with the final community partition stored as a label vector or dictionary mapping nodes to communities.

Extension Considerations: Newman's algorithm has high computational complexity (O(n³) for naive implementations), making it suitable for small to medium-sized networks. Local optima in modularity optimization may lead to unstable partitions, which can be improved by incorporating multi-resolution or spectral methods. The standard division of Zachary network contains 2 communities, consistent with the actual club分裂 event, allowing for algorithm validation through precision_recall metrics or NMI scores.