connectivity analysis

Manual exploration in graph theory research is inherently limited and lacks systematic rigor. Method: We propose the first scalable, computer-assisted research framework integrating mixed-integer linear programming, semidefinite programming, SAT solving, metaheuristic algorithms, and machine learning. Our approach combines graph isomorphism enumeration, construction of searchable graph databases, dynamic programming, and algebraic computation to enable complete generation of graphs within specified classes and efficient invariant analysis. Contribution/Results: This work establishes the first unified formalism for synergistic application of diverse algorithmic paradigms in graph theory. It automatically discovers novel conjectures and counterexamples in extremal graph theory, graph coloring, and spectral graph theory. Empirical evaluation confirms the framework’s dual advantages—enhanced computational efficiency and deeper theoretical insight—thereby significantly advancing automation and reproducibility in graph-theoretic research.

This work addresses the representation learning challenge for higher-order networks (e.g., hypergraphs). We propose a unified spectral-moment-based framework: higher-order graphs are decomposed by edge order into uniform hypergraphs; order-specific random walk transition matrices are defined; and low-order moments of their spectral densities are extracted as joint features. This constitutes the first systematic extension of spectral moment theory to higher-order networks, establishing analytical connections between spectral moments and higher-order structural properties—such as higher-order degree distributions and hyperedge clustering coefficients—thereby overcoming the expressive limitations of conventional pairwise graph spectral methods. By fusing multi-order spectral moments, our representation achieves significant performance gains over state-of-the-art methods on higher-order graph classification tasks. Experiments demonstrate that the proposed representation accurately captures walk return probabilities and diverse higher-order topological characteristics.

Novice learners in network science lack systematic, interdisciplinary guidance. Method: We propose the “Network Scientist Training Map” framework, integrating diverse methodological tools—including graph theory, random graph models, spectral graph theory, network embedding, statistical inference, dynamical modeling, and interpretable machine learning—restructuring core paradigms in an accessible, unified manner that emphasizes methodological synthesis over technical accumulation. Contribution/Results: First, we formally establish pure network science as an autonomous academic discipline. Second, we construct a structured, full-stack knowledge graph covering foundational concepts and techniques. Third, by transcending the limitations of single-discipline textbooks, the framework enables learners with no prior background to rapidly develop a coherent, systems-level conceptual framework—thereby advancing network science from an application-oriented auxiliary field toward ontological independence.

Multi-partite network spectral embeddings reside in high-dimensional spaces, yet their intrinsic node representations lie within group-specific low-dimensional subspaces—a geometric structure previously uncharacterized. Method: This paper introduces the first post-processing dimensionality reduction method with theoretical consistency guarantees to recover the intrinsic dimensionality of such embeddings. Grounded in a low-rank inhomogeneous random graph model, the method jointly leverages subspace estimation and matrix perturbation theory to provably achieve consistent subspace recovery; it further unifies and generalizes the bipartite spectral embedding framework. Results: Extensive experiments demonstrate that the proposed method significantly outperforms standard spectral embedding and conventional bipartite embedding approaches on clustering and visualization tasks, achieving both theoretical rigor and practical effectiveness.

This work investigates the discriminative power of spectral angles—the angles between eigenvectors of the adjacency matrix and standard basis vectors—for graph isomorphism testing. Methodologically, it establishes a purely combinatorial characterization of spectral angles at the level of walk counts, enabling a precise comparison with the Weisfeiler–Leman (WL) hierarchy. The contributions are threefold: (i) it proves that spectral angles are strictly equivalent in expressive power to the 2-dimensional WL algorithm (2-WL), yet strictly weaker than 3-WL—thereby fully resolving an open problem posed by Fürer regarding this invariant; (ii) it uncovers intrinsic connections between spectral angles, generalized spectra, and principal spectra; and (iii) it demonstrates that “almost all graphs are uniquely determined by their spectrum together with spectral angles”, yielding significant progress toward the long-standing conjecture on spectral uniqueness of graphs.

This study addresses the spectral determination of graphs and the edge- and distance-transitivity of generalized Hamming graphs and their complements. By integrating tools from spectral graph theory, group actions, and association schemes—including the Cauchy interlacing theorem, Schur complements, and Cayley graph constructions—the work introduces a novel family of “pyramid graphs” and establishes their adjacency-spectral uniqueness. It fully characterizes the conditions under which generalized Hamming graphs and their complements are edge-transitive or distance-transitive, and derives closed-form expressions for their Lovász ϑ-functions. The results yield new spectral characterizations of complete bipartite graphs, Turán graphs, and strongly regular graphs, confirm the spectral uniqueness of pyramid graphs, and systematically uncover the intrinsic connections between the symmetry and spectral properties of generalized Hamming graphs.

This work addresses the limitations of traditional graph signal processing, which is confined to node-level signals and thus unable to capture higher-order interactions inherent in complex systems. By leveraging simplicial complexes and combinatorial Hodge Laplacians, the study extends signal processing to higher-dimensional topological structures such as edges and triangles. It introduces a method for constructing higher-order signals from lagged node observations and develops a corresponding theory of topological Fourier transforms and filtering. Applied to brain imaging data, the proposed framework successfully uncovers nontrivial higher-order interaction patterns among sets of brain regions that are invisible to conventional approaches, thereby establishing a theoretical and practical bridge for higher-order topological signal processing.

Traditional graph representations face significant challenges in graph isomorphism testing and symmetry-aware visualization due to high computational complexity and low efficiency. This work proposes “graph linear notation”—a complete graph invariant derived from canonical form algorithms—and establishes it, for the first time, as an equivalent definition for finite graphs. This representation not only substantially simplifies graph isomorphism comparison and symmetry-aware visualization but also naturally accommodates the extension and application of classical graph-theoretic concepts, such as coloring and paths, within its framework. By unifying these capabilities, the proposed notation offers a highly efficient and coherent new paradigm for structural graph analysis.

This work addresses the lack of a unified and reproducible evaluation framework in existing hyperbolic graph representation learning methods, which hinders systematic comparison and practical deployment. We propose an open-source, standardized framework that integrates multiple state-of-the-art hyperbolic graph embedding algorithms, offering consistent training pipelines, visualization tools, and evaluation interfaces for downstream tasks such as link prediction and node classification. The framework seamlessly interoperates with widely used network analysis libraries. Comprehensive experiments on real-world networks not only validate the predictive performance of various methods but also uncover their respective strengths and limitations, thereby providing empirical guidance for method selection. This significantly enhances reproducibility and practical efficiency in hyperbolic graph learning research.

Traditional graph representation learning treats the latent dimension as a fixed hyperparameter, which inadequately captures the model’s true capacity and suffers from non-identifiability due to rotational and scaling ambiguities in latent factors. This work proposes Spectra, a method that uses the spectral distribution of normalized positive-definite kernels as the fundamental analytical unit. By leveraging Shannon effective rank as a dynamic measure to quantify and control model capacity, Spectra reinterprets capacity not as an external hyperparameter but as an intrinsic property of the model itself. Through spectral prefix extraction, trace-normalized kernel matrices, and bisection-based optimization, Spectra uncovers performance–capacity trade-offs across diverse real-world networks, achieves competitive results against strong baselines in link prediction, and enables the generation of aligned low-dimensional views at varying capacities from a single trained model.