🤖 AI Summary
Existing graph sparsification methods struggle to preserve multiscale connectivity structures under edge relevance filtering, often discarding topological evidence critical for community detection. This work proposes a persistence-aware sparsification approach that, given an edge relevance filtration and a proxy partition, selects a budget-constrained subgraph via constrained greedy submodular optimization to exactly maintain connected components across all filtration thresholds—thereby achieving, for the first time, complete preservation of the zeroth-dimensional persistent diagram with a theoretical $(1 - 1/e)$ approximation guarantee. The method integrates a persistent homology skeleton, a skeleton-constrained monotone submodular objective, and a degree-balanced recovery strategy. Experiments on six real-world web and social graphs demonstrate that TopoBudget achieves state-of-the-art community preservation under Louvain, competitive performance under Infomap, zero topological mismatch, and significantly faster runtime than effective-resistance baselines.
📝 Abstract
Web and social graphs are analyzed repeatedly for community structure, yet many of their edges are redundant for this purpose, which motivates sparsification. Existing sparsifiers preserve spectral quantities, cuts, local similarity, or a single clustering, but none preserves the thresholded connectivity structure of an edge-relevance filtration, the multiscale pattern by which groups form at high relevance and merge through weaker bridges. We study persistent-connectivity-preserving sparsification: given a graph, an edge-relevance filtration, and a proxy partition computed once during preprocessing, select a budgeted subgraph that preserves the labeled component partition at every threshold, and hence the zero-dimensional persistence diagram, while retaining community evidence for later analyses. Our method, TopoBudget, first extracts a tie-aware persistence backbone that enforces this constraint, then allocates the residual edge budget by greedily maximizing a backbone-conditioned submodular objective that rewards balanced recovery of proxy-internal degree. We prove exact preservation of the component partition at every threshold, and that the conditioned objective is monotone and submodular, so greedy attains a (1-1/e) guarantee for the fixed-backbone residual problem. On held-out synthetic benchmarks and six real Web and social graphs at equal budget, TopoBudget gives the strongest community preservation among topology-preserving methods under Louvain, remains competitive under Infomap, incurs zero topology mismatch, and runs substantially faster than an effective-resistance baseline. A no-backbone ablation shows that, on the real graphs, the mandatory backbone improves average quality while providing the exact guarantee. TopoBudget thus couples exact multiscale connectivity with budgeted, reusable community preservation.