cycle detection

Detecting the minimum-weight cycle (MWC) in weighted graphs suffers from high computational complexity and a lack of efficient deterministic algorithms. To address this, we propose a novel node-iterative optimization algorithm grounded in the Dijkstra framework. Our key contributions are: (1) a composite distance metric integrating path weight and cycle-closure cost; (2) a provably correct node pruning criterion leveraging loop invariants for dynamic elimination of suboptimal intermediate results; and (3) a local-structure-aware subgraph pruning heuristic. The algorithm guarantees global optimality while significantly reducing time complexity. Extensive experiments on diverse real-world graphs demonstrate speedups of 1.5×–5.8× over state-of-the-art methods. Moreover, it enables efficient computation of higher-order network features such as Loop Modulus, underscoring its practical utility in large-scale graph analytics.

This work addresses the challenge of identifying and defining typical graph patterns—such as cliques and bicliques—in real-world graph data corrupted by noise. To this end, the authors propose a noise-robust approach based on adjacency matrix reordering and Moran’s I statistic. The method first employs Moran’s I to derive an optimal node ordering that transforms latent graph patterns into well-structured rectangular submatrices. It then combines exact algorithms with heuristic strategies to efficiently decompose noisy patterns. Furthermore, a noise-aware motif simplification technique is introduced to facilitate effective visualization. Extensive experiments on multiple real-world datasets demonstrate that the proposed method reliably identifies and visualizes graph patterns under noisy conditions, confirming its effectiveness and practical utility.

This work addresses the lack of principled, quantitative evaluation criteria for schema-noise decomposition quality in graph data. We propose SCHENO, the first theoretically grounded and interpretable evaluation framework for this task. SCHENO integrates information-theoretic principles with graph reconstruction error to jointly quantify decomposition quality along three orthogonal dimensions: schema regularity, noise randomness, and overall fidelity. Methodologically, it employs a lightweight optimization strategy for efficient schema discovery and supports visual attribution on small graphs for interpretability and validation. Experiments demonstrate that SCHENO effectively guides the discovery of diverse structural patterns; for the first time, it quantitatively exposes systematic suboptimality in mainstream graph mining algorithms (e.g., gSpan, GSpan) regarding schema representation; and it establishes a reproducible, comparable benchmark for evaluating graph decomposition methods. (149 words)

This paper addresses root cause localization of outliers in cyclic causal graphs, focusing on outlier propagation mechanisms under linear structural equation models (SEMs). We propose a structure-agnostic root cause identification method that reconstructs latent anomaly sources directly from observational data, by modeling perturbation propagation paths within system dynamics and integrating causal inference with graph-theoretic analysis. Our key contribution is extending root cause localization to cyclic causal graphs with unknown topology; under a strong perturbation assumption, we prove that the algorithm outputs a short candidate set containing—w.h.p.—both the true root cause and its immediate parents along the causal cycle. Experiments demonstrate the framework’s feasibility and accuracy on systems with unknown cyclic structures, significantly enhancing interpretability and practicality of anomaly diagnosis in complex feedback systems.

The computational complexity of graph isomorphism testing remains unresolved, particularly for highly symmetric graphs whose adjacency matrices possess repeated eigenvalues—causing ambiguity in the solution space for conventional methods. This paper introduces a novel continuous optimization framework: it reformulates the discrete matching problem via orthogonal and doubly stochastic relaxations, and—crucially—systematically characterizes how eigenvalue multiplicity governs the geometric structure of the feasible solution space. Building on this insight, we propose a subspace constraint strategy that effectively suppresses spurious solutions induced by symmetry. Our algorithm employs the Frank–Wolfe method, integrating spectral matrix analysis with iterative projection-based optimization. Extensive evaluation on highly symmetric benchmarks—including strongly regular graphs, complete graphs, and the Petersen graph—demonstrates substantial improvements in both efficiency and robustness of isomorphism detection.

This paper addresses the problem of identifying and separating multicomponent signals in time-frequency (TF) representations. We propose a novel graph-clustering paradigm: the TF structure of a signal is modeled as a weighted graph, where nodes represent TF bins and edge weights encode local similarity; spectral clustering or graph partitioning algorithms are then applied to automatically detect well-connected subgraphs with minimal inter-cluster connections—each corresponding to an individual signal component. Our key contribution is the first rigorous establishment of an equivalence between TF signal decomposition and graph clustering, yielding an interpretable and scalable graph-theoretic framework for signal separation. Numerical experiments demonstrate that the method achieves robust separation under challenging conditions—including additive noise, nonstationarity, and severe TF overlap—outperforming conventional TF filtering approaches.

Existing inconsistency thresholds for incomplete pairwise comparison matrices—containing unknown comparisons—lack interpretability, as their design ignores the underlying graph structure formed by known comparisons. Method: We establish that the threshold depends not merely on matrix size and number of missing entries, but fundamentally on the topology of the associated undirected comparison graph; we derive a quantitative relationship between the threshold and the graph’s spectral radius, and propose a dynamic inconsistency criterion grounded in spectral graph properties. Leveraging graph-theoretic modeling, spectral analysis, large-scale statistical simulation, and inconsistency proposition theory, we construct a real-time, embeddable threshold framework for decision support systems. Contribution/Results: The proposed method enables online anomaly detection during data collection and significantly enhances the robustness and reliability of multi-criteria decision making under incompleteness.

Addressing the core challenges in persistent homology—namely, cycle feature identification, cross-dataset cycle matching, and cycle community construction—this paper proposes a dual-path framework. First, it introduces a novel Wasserstein distance metric integrated with merge trees to construct discriminative cycle dendrogram representations. Second, it designs Stratified Gradient Sampling—a hierarchical, multi-filter function co-learning strategy—that enables non-overlapping, exhaustive partitioning of cycle communities. By optimizing cycle centroid functions and generating topology-driven filter functions, the model achieves precise reconstruction of multiple independent cycles and structural alignment across objects on synthetic data. The framework significantly enhances interpretability and transferability of the cycle space, enabling robust topological analysis across diverse datasets while preserving geometric and topological fidelity.

This work addresses the problem of efficiently approximating the count of 4-cycles in graph edge streams presented in arbitrary order. To this end, the authors propose two induced subgraph sampling–based algorithms: a two-pass algorithm that achieves theoretically optimal space complexity on graphs with bounded degeneracy, marking the first optimal solution in this streaming model; and a single-pass algorithm tailored for sparse networks where 4-cycles are uniformly distributed, demonstrating robust performance on non-bipartite graphs such as social networks. Experimental evaluation shows that the two-pass algorithm significantly outperforms existing methods on real-world graph streams, while the single-pass variant also exhibits strong performance in its intended application scenarios.

This work investigates the query complexity lower bound for cycle detection in colored grid graphs, where edges are defined by adjacent cells of the same color. For any $m \times n$ grid with $m, n \geq 2$, we establish a tight $\Omega(mn)$ lower bound by proving that every deterministic algorithm must query all $mn$ cells in the worst case. The core technical contribution is a novel adversarial construction based on block-wise opposition and checkerboard isolation. By composing local adversaries of sizes $2\times2$, $2\times3$, $3\times2$, and $3\times3$, an adaptive adversary can maintain ambiguity—preserving both cyclic and acyclic colorings—until the algorithm queries the very last cell.