Systematic evaluation of a new combinatorial curvature for complex networks
Abstract: We have recently introduced Forman's discretization of Ricci curvature to the realm of complex networks. Forman curvature is an edge-based measure whose mathematical definition elegantly encapsulates the weights of nodes and edges in a complex network. In this contribution, we perform a comparative analysis of Forman curvature with other edge-based measures such as edge betweenness, embeddedness and dispersion in diverse model and real networks. We find that Forman curvature in comparison to embeddedness or dispersion is a better indicator of the importance of an edge for the large-scale connectivity of complex networks. Based on the definition of the Forman curvature of edges, there are two natural ways to define the Forman curvature of nodes in a network. In this contribution, we also examine these two possible definitions of Forman curvature of nodes in diverse model and real networks. Based on our empirical analysis, we find that in practice the unnormalized definition of the Forman curvature of nodes with the choice of combinatorial node weights is a better indicator of the importance of nodes in complex networks.
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Knowledge Gaps
Knowledge gaps, limitations, and open questions
Below is a single consolidated list of what remains missing, uncertain, or unexplored in the paper, articulated to enable concrete follow-up by future researchers.
- Formal treatment and empirical evaluation of Forman curvature on directed, signed, temporal, multiplex, and multigraph networks; clear definitions and algorithms for these settings are absent.
- Systematic analysis of edge weights: although the method admits arbitrary positive weights, experiments were limited to (mostly) unweighted graphs and two node-weight schemes; the impact of realistic edge weights (distance, capacity, strength, inverse-distance) on curvature and connectivity was not tested.
- Broader node-weight choices beyond 1 and degree (e.g., weighted degree/strength, attribute- or domain-informed weights, learned weights) were not explored, nor was theoretical guidance provided for choosing node weights in different network classes.
- Higher-dimensional Forman curvature in simplicial complexes was not leveraged (e.g., curvature on faces/triangles/tetrahedra in NGF), leaving open whether curvature on higher-dimensional cells yields stronger signals for robustness or structure.
- No direct, controlled benchmarking against Ollivier-Ricci curvature across identical datasets, metrics, and computational budgets; the complementary or redundant value of Forman vs. Ollivier curvature is unclear.
- Lack of theoretical characterization of Forman curvature distributions: no analytical expectations, bounds, or asymptotics for ER/WS/BA/NGF models; no proofs linking curvature to spectral properties (e.g., Laplacian eigenvalues, algebraic connectivity) or percolation thresholds.
- Degree confounding in node curvature analyses: unnormalized node curvature strongly correlates with degree; partial correlations, regression models, or degree-normalized variants to isolate independent signal were not performed.
- Absence of principled normalization schemes for node curvature beyond averaging by degree; alternatives (variance-stabilizing, residualization against degree/strength, scaling by local volume) were not examined.
- Scalability and complexity: no empirical runtime, memory, or parallelization analysis on large-scale graphs (≥106 nodes); comparative cost vs. Brandes betweenness and other edge metrics remains unquantified.
- Robustness to noise and sampling: sensitivity of curvature to missing/spurious edges, sampling biases, and measurement errors was not assessed.
- Edge removal protocol ambiguity: it is unclear whether rankings were recomputed after each removal (adaptive attack) or fixed from the initial graph; the effect of recalculating curvature/centrality during attacks remains unexplored.
- Validation metrics for “importance” of edges/nodes were limited to communication efficiency; alternative global metrics (size of largest connected component, average path length, algebraic connectivity, assortativity, giant-component percolation threshold) were not evaluated.
- Parameter-space coverage is narrow: the analysis did not sweep broad ranges for ER p, WS k/β, BA m, NGF flavor s and dimension d; generalizability across model parameter regimes is uncertain.
- Real weighted networks were not analyzed using their native weights (e.g., flow capacities in power grids, interaction strengths in protein networks); mapping domain-specific weights into curvature and their effects remain open.
- Interpretability of curvature extremes: the local structural motifs (e.g., bridges, wedges, triangles, k-core boundaries) that produce highly negative or positive Forman curvature were not systematically characterized.
- Hybrid predictors: combining Forman curvature with global measures (betweenness), local measures (embeddedness, edge clustering), and spectral features was not tested for improved edge/node importance prediction.
- Dynamics: how curvature evolves over time in growing or temporal networks, and whether it predicts emerging hubs or bridges, was not studied.
- Statistical reporting: correlations and attack curves lack confidence intervals, hypothesis tests, and effect-size measures; statistical significance and reproducibility of observed differences are unclear.
- Normalized edge curvature variants (analogous to node normalization) were not proposed or evaluated to mitigate scale or degree effects at the edge level.
- Reproducibility: code, parameter settings, and pipelines were not documented for independent replication; sensitivity of results to implementation choices is unknown.
- Applied tasks: utility of Forman curvature for community detection, link prediction, anomaly detection, and flow routing was not evaluated.
- Signed weights: extension of Forman curvature to networks with negative edge weights (e.g., inhibitory interactions) and the requisite mathematical handling was not addressed.
- Conditions under which Forman curvature outperforms betweenness: beyond NGF with explicit geometry, structural diagnostics (e.g., hyperbolicity, latent geometric embedding) that predict superiority of curvature were not identified.
- Formal redundancy or equivalence of degree as a node weight: the paper hypothesizes redundancy, but provides no proof or systematic counterexamples; conditions under which node-degree weighting changes curvature outcomes remain open.
- Edge–node curvature interplay: whether node curvature adds predictive power for edge importance (and vice versa) beyond degree/centrality in multivariate models was not investigated.
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