GN Algorithm in Complex Networks: Community Detection Implementation

The Girvan-Newman (GN) algorithm is a classical community detection algorithm in complex network analysis, designed to identify community structures within networks. This algorithm operates based on the concept of edge "betweenness centrality," progressively removing the most critical edges to partition the network into distinct communities. In code implementation, this typically involves calculating shortest paths between all node pairs using algorithms like Brandes' betweenness centrality.

The core principle of the GN algorithm states that edges connecting different communities generally exhibit high betweenness centrality (meaning these edges lie on the shortest paths between many node pairs). By iteratively computing and removing edges with the highest betweenness values, the algorithm gradually decomposes the network into separate community structures. From a programming perspective, this requires maintaining dynamic data structures to track edge removals and recompute betweenness after each iteration.

A crucial evaluation metric for the GN algorithm is Modularity, which quantifies the quality of community partitions. Higher modularity indicates stronger connection density within communities and sparser connections between communities. The algorithm optimizes modularity to determine the optimal community partition scheme. Implementation-wise, modularity calculation involves comparing actual edge distributions with expected random distributions using Q = (1/2m)Σ[ A_ij - k_ik_j/2m ] δ(c_i,c_j) where m represents total edges and A_ij is the adjacency matrix.

Although the GN algorithm has high computational complexity (O(m²n) for n nodes and m edges), its clear logic and effectiveness maintain its widespread application in social networks, biological networks, and transportation networks. Many subsequent community detection algorithms (like the Louvain method) draw inspiration from GN's approach while improving computational efficiency through greedy optimization and hierarchical aggregation techniques.