graph metric analysis

This work systematically investigates the theoretical foundations and computational feasibility of graph similarity measures. Addressing mainstream graph distance definitions—including graph edit distance, spectral distance, and subgraph matching—the paper establishes, for the first time, a unified mathematical characterization of their essential properties and applicability boundaries, thereby clarifying intrinsic connections among spectral, combinatorial, and learning-based approaches. Leveraging rigorous tools from graph edit distance theory, Laplacian spectral analysis, subgraph isomorphism testing, and computational complexity theory, the study precisely delineates the computability boundaries of these distances, identifies the fundamental sources of their NP-hardness, and characterizes conditions under which efficient approximation is feasible. The results provide a principled theoretical framework for selecting appropriate graph similarity algorithms and prescribe scalable approximate computation strategies for large-scale graphs—bridging deep theoretical insight with practical algorithmic guidance.

Quantifying similarity among graph-structured data remains challenging due to the lack of statistically rigorous, interpretable metrics. Method: This paper proposes a network similarity assessment framework grounded in statistical hypothesis testing. It constructs a detection framework sensitive to subtle structural perturbations and integrates multi-scale topological features—including degree distribution, clustering coefficient, and average path length—leveraging asymptotic distribution theory for significance inference. Contribution/Results: Extending prior theoretical guarantees, the method is rigorously evaluated across dozens of synthetic graph families and real-world networks (social, biological, infrastructure). It significantly improves detection accuracy for graph isomorphism-preserving perturbations, sampling bias, and generative model mismatch. Experiments demonstrate high statistical power, strong robustness to noise and structural heterogeneity, and cross-domain generalizability. The approach provides an interpretable, reproducible benchmark for evaluating graph generative models, analyzing network evolution, and validating structural equivalence in complex systems.

This work addresses the challenge of defining distance metrics between heterogeneous, large-scale graphs. We propose a family of Generalized Optimal Subpattern Assignment (GOSPA) distances for graphs, the first to rigorously satisfy all four metric axioms—non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Our formulation unifies node attribute dissimilarity, penalties for unmatched nodes, and a tunable, parameterized cost for edge-structure mismatches. The resulting distance is efficiently approximated via linear programming, ensuring both theoretical soundness and computational tractability. Extensive evaluation on multiple synthetic and real-world graph datasets demonstrates that our GOSPA-based metric significantly outperforms conventional graph distance measures—including Graph Edit Distance and Frobenius norm-based distances—in classification tasks. Empirical results confirm its enhanced structural sensitivity and discriminative power, establishing it as a principled and practical tool for graph comparison in heterogeneous settings.

Existing hypergraph clustering coefficients treat hyperedges as atomic units, ignoring pairwise interactions among their constituent nodes—leading to spurious zero values for nodes embedded in nontrivial clustering structures. Method: We propose a novel hypergraph clustering coefficient that explicitly models intra-hyperedge pairwise relational strength via a mapping from hypergraphs to weighted graphs. Contribution/Results: The proposed coefficient rigorously satisfies three theoretical desiderata: (i) boundedness in [0,1], (ii) consistency with the classical graph clustering coefficient upon graph degeneration, and (iii) faithful characterization of higher-order local structure. Validated through higher-order motif analysis and real-world social and collaboration datasets, it significantly corrects the zero-value bias of conventional methods on 3-node motifs (III, IV-a, IV-b) and provides finer-grained, more accurate quantification of local density—especially for large hyperedges.

Existing axiomatic studies of centrality primarily focus on isolated or few measures, lacking a unified characterization of the commonalities and distinctions among feedback-based centralities. This paper introduces the first general axiomatic framework encompassing four canonical feedback centralities: eigenvector centrality, Katz centrality, Katz prestige, and PageRank. Leveraging axiomatic analysis, graph theory, and linear algebra, we rigorously prove that each centrality is uniquely characterized by a minimal complete subset of this framework. Our analysis reveals their fundamental similarities and differences in normalization schemes, diffusion mechanisms, and boundary condition handling. Moreover, the framework establishes a theoretical foundation for interpretability and cross-measure comparison of centrality measures. To our knowledge, this is the first systematic axiomatic framework supporting principled modeling and selection of centrality measures in network science.

Existing graph centrality measures lack a unified, quantifiable framework for systematic comparison, hindering the formalization and validation of related conjectures. This work proposes a mathematical approach based on vertex rankings to construct the first computable approximation framework capable of systematically comparing any two centrality measures. By integrating graph theory, formal modeling, and approximation algorithms, the method not only verifies several classical conjectures but also generates novel hypotheses of independent research interest. The framework thus establishes a theoretical foundation for network science and opens new avenues for future investigation.

This study addresses the problem of effectively measuring node centrality in graphs from geometric and topological perspectives. To this end, it introduces magnitude homology—a novel application in graph centrality analysis—and proposes a local centrality measure grounded in relative homology: the importance of a node is quantified by the change in magnitude homology resulting from its removal. The proposed measure satisfies several natural axioms, exhibits favorable theoretical properties, and demonstrates unique effectiveness in experiments, offering complementary insights to classical centrality metrics. This work thus provides a new topological lens for evaluating node importance in complex networks.

Existing methods for comparing graph partitions often disregard the underlying graph topology, making it difficult to accurately capture the cohesion within and separation between communities. This work proposes a graph-aware distance framework that constructs a topology-respecting metric by inducing edge partitions, enabling meaningful comparison of continuous graph partitions. The framework adheres to a local graph-aware refinement criterion and is theoretically shown—under the stochastic block model—to be sensitive to topological perturbations. Specifically, the proposed distance almost surely increases with stronger perturbations in both intra- and inter-community splitting scenarios, significantly outperforming conventional metrics such as variation of information, van Dongen distance, and binary cut distance. This provides a structurally consistent and topology-sensitive criterion for evaluating graph partitions.

This paper addresses the problem of attribute-based node comparison in attributed graphs—a task largely overlooked by prior work, which focuses predominantly on node importance scoring rather than automated extraction of discriminative insights. We formalize two core problems: (i) constructing interpretable, attribute-aware comparison metrics, and (ii) grouping nodes by statistical significance of their differences. To solve them, we propose a multi-level heuristic framework integrating context-aware metric generation, combinatorial optimization modeling, and multi-strategy search—balancing computational efficiency with comparative depth and interpretability. Extensive evaluation on real-world attributed graph datasets demonstrates that our lightweight variant delivers actionable insights within minutes, while the high-fidelity version substantially improves granularity and semantic coherence of comparisons. To the best of our knowledge, this is the first scalable, interactive, and fully interpretable automation framework for attribute-driven node comparison in attributed graphs.

This study addresses the limitations of traditional node similarity measures, which often assume a uniform and continuous feature space and thus fail to capture the true structural equivalence among nodes in attributed networks. By integrating neighborhood attribute profiling, dimensionality reduction, and visualization techniques, the authors uncover complex nonlinear manifold structures and density biases inherent in high-dimensional feature spaces. Empirical analysis on an enterprise transaction network reveals that semantically identical industry labels can correspond to multiple disconnected regions of structural roles, and that supply chain tiers exhibit continuous transitions rather than discrete partitions. These findings motivate the proposal of a new similarity metric grounded in manifold topology to more accurately reflect structural equivalence among nodes.