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variants of the algorithm were tested against a benchmark suite of 1,200 synthetic graphs, each generated with a distinct combination of density parameters, degree distributions, and community structures. The first variant, which used a simple greedy heuristic for initial vertex selection, consistently produced partitions with modularity scores between 0.62 and 0.71 on graphs with high clustering coefficients, but its performance degraded sharply on sparse graphs with power-law degree distributions, where it often got stuck in local optima and left many low-degree vertices isolated. The second variant introduced a randomized restart mechanism that diversified the search space by perturbing the seed set after every 50 iterations; this improved the worst-case modularity by 0.08 on average, but at the cost of a 2.3x increase in runtime, making it impractical for graphs exceeding 500,000 edges. The third variant, which incorporated a hierarchical multi-level refinement step, showed the most consistent behavior: it achieved modularity scores within 0.03 of the best-known values across all graph types, and its memory footprint remained under 4 GB even for the largest test case, a 2-million-vertex graph with 18 million edges. However, this variant required careful tuning of the coarsening ratio and the refinement threshold, and its performance was sensitive to the initial ordering of vertices, which introduced a variance of up to 0.015 in modularity across different runs on identical inputs. To address this, we implemented a fourth variant that used a deterministic ordering based on a breadth-first traversal from a pseudo-random root, which eliminated the variance but reduced the peak modularity by 0.01 on graphs with strong community structure. The fifth variant combined the randomized restart with the hierarchical refinement, and this hybrid approach outperformed all others on graphs with overlapping communities, achieving modularity scores up to 0.79, but it required a careful balance between the restart frequency and the refinement depth; setting the restart interval too low caused the algorithm to revisit similar partitions, while setting it too high caused it to miss high-quality solutions. We also tested a parallel implementation of the third variant using OpenMP with 16 threads, which showed near-linear speedup up to 8 threads, but the speedup plateaued beyond that due to synchronization overhead in the refinement phase. On the memory side, the fifth variant used a compressed sparse row format for the adjacency list, which reduced memory consumption by 40% compared to the adjacency list representation, but this came at the cost of a 15% slowdown in the coarsening step. We then evaluated all variants on a real-world dataset comprising 45 social networks and 12 citation networks, where the third variant consistently produced the most balanced partitions in terms of community size distribution, while the fifth variant excelled at detecting small, dense communities that were missed by other methods. For the social networks, the average modularity improvement over the baseline Louvain method was 0.04 for the third variant and 0.06 for the fifth variant, but the fifth variant took 3.5 times longer on the largest network, which had 3.2 million nodes and 45 million edges. We also analyzed the sensitivity of each variant to the resolution parameter gamma, which controls the granularity of the communities; the first variant was highly sensitive, with modularity dropping by 0.15 when gamma was increased from 1.0 to 1.5, while the third variant showed a more gradual decline of 0.05 over the same range. The fourth variant was the least sensitive, but it produced overly coarse partitions at low gamma values, missing many small communities. To further stress-test the variants, we generated graphs with planted partitions of varying sizes and strengths, and we measured the normalized mutual information (NMI) between the detected and ground-truth communities. The third variant achieved an NMI of 0.93 on strong partitions, but this fell to 0.71 on weak partitions where the inter-community edge probability was only 0.02 higher than the intra-community probability. The fifth variant performed better on weak partitions, achieving an NMI of 0.82, but it was slower by a factor of 2.8. We also examined the scalability of the variants on graphs with up to 10 million vertices, where the first variant ran out of memory due to its dense intermediate data structures, while the third and fifth variants completed within 12 and 25 minutes respectively. The second variant was not tested beyond 2 million vertices because its restart mechanism caused excessive I/O overhead. Finally, we compared the variants on graphs with weighted edges, where the edge weights followed a log-normal distribution; the third variant handled weighted graphs gracefully, but the first variant treated weights as binary, leading to a 0.12 drop in modularity. The fifth variant, when adapted to use weighted modularity, showed a modest improvement of 0.02 over its unweighted counterpart, but it required a 10% increase in runtime. Overall, the third variant is recommended for general use due to its robustness and efficiency, while the fifth variant is preferred for applications requiring the detection of small, dense communities or when working with graphs that have overlapping community structures. We also note that all variants assume an undirected graph; for directed graphs, the modularity definition must be modified, and preliminary tests on a directed citation network showed that the third variant, when adapted for directed modularity, produced partitions with a higher conductance but lower modularity compared to the undirected case, indicating a trade-off that needs further investigation. In terms of implementation details, all variants were written in C++ and compiled with -O3 optimization, and the source code is available under an open-source license. The benchmark suite, including the graph generator and the ground-truth partitions, is also publicly available for reproducibility. We believe that the systematic comparison presented here provides a clear guide for practitioners choosing a community detection algorithm, and the insights on parameter sensitivity and scalability will inform future algorithmic developments.

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