Efficient modularity optimization by multistep greedy algorithm and vertex mover refinement (0712.1163v2)

Abstract: Identifying strongly connected substructures in large networks provides insight into their coarse-grained organization. Several approaches based on the optimization of a quality function, e.g., the modularity, have been proposed. We present here a multistep extension of the greedy algorithm (MSG) that allows the merging of more than one pair of communities at each iteration step. The essential idea is to prevent the premature condensation into few large communities. Upon convergence of the MSG a simple refinement procedure called "vertex mover" (VM) is used for reassigning vertices to neighboring communities to improve the final modularity value. With an appropriate choice of the step width, the combined MSG-VM algorithm is able to find solutions of higher modularity than those reported previously. The multistep extension does not alter the scaling of computational cost of the greedy algorithm.

Modularity Optimization in Networks via Multistep Greedy Algorithm and Vertex Mover Refinement

The paper by Philipp Schuetz and Amedeo Caflisch addresses the challenge of optimizing modularity in networks by introducing an advancement in algorithmic strategies. The authors propose a multistep greedy algorithm (MSG) integrated with a vertex mover refinement (VM) to enhance community detection in large-scale networks. This technique is directed at improving the partitions of networks into communities such that the modularity—defined as the ratio of intra-community edges compared to what would be expected in a random network—is maximized.

Algorithm Overview

The MSG variant elevates traditional greedy algorithms by allowing simultaneous merging of multiple pairs of communities at each iteration step, circumventing the premature formation of oversized communities. The amalgamation process is guided by two criteria: selecting community pairs whose modularity change, $\Delta Q$, is among the most favorable values and ensuring that the merging does not involve communities already modified for more significant modularity changes in the current iteration. This procedure is designed to uphold efficiency in terms of computational cost, which remains consistent with classical methods.

Upon completion of the MSG process, a local refinement strategy termed vertex mover (VM) is employed. VM iteratively reassigns vertices to adjacent communities to achieve a higher modularity by examining potential improvements in the reallocation of individual vertices. This method shares similarities with the Kernighan-Lin algorithm but focuses purely on local optimizations without repelling modularity-enhancing reassignments.

Numerical Results and Comparison

The efficacy of the MSG-VM approach is underscored by benchmarking against both classical greedy methods and previously reported modularity values on standard network datasets, including social networks, metabolic networks, and collaboration graphs. For instance, the MSG-VM algorithm reported higher modularity values than extant techniques in the collaboration network and the PGP-key signing network, among others. However, it falls short in the Zachary Karate Club network when compared to the Kernighan-Lin derived optimal modularity values.

Implementation and Computational Complexity

Implementing the MSG encompasses efficient recalculation of modularity changes for affected community pairs using sparsely stored vectors for connectivity matrices, ensuring quick access and updates. The computational complexity of MSG is predicted to be $O(DM \log(N))$, where $D$ is the depth of the community dendrogram, $M$ is the number of edges, and $N$ is the number of vertices. This complexity is akin to conventional greedy algorithms. Meanwhile, the VM process benefits from streamlined memory access to vertex community affiliation, resulting in $O(L)$ time complexity per iteration.

Implications and Future Directions

The MSG-VM technique offers a more scalable solution for enhancing modularity in large networks, critical for domains like systems biology, social network analysis, and web structure evaluation. The approach provides insights into understanding the inherent modular structures across complex systems.

However, future research may explore automated strategies for dynamically adjusting the multistep parameter $l$ and explore extending the method's applicability to directed or weighted networks. Moreover, examining the algorithm's performance across synthetic networks and contemplating its integration with other optimization techniques may yield further advancements.

In summary, the proposed strategy marks a significant stride in modularity-based community detection algorithms, fostering efficient partitioning that captures high-fidelity representations of network structures.