Search Results (53)

Large language models (LLMs) have driven transformative progress in artificial intelligence, yet critical challenges persist in data management and privacy protection during model deployment and training. The approximate nearest neighbor (ANN) search, a core operation in LLMs, faces inherent trade-offs between efficiency and security when implemented through conventional locality-sensitive hashing (LSH)-based secure ANN (SANN) methods, which often compromise either query accuracy due to false positives. To address these limitations, this paper proposes a novel secure ANN scheme based on nearest neighbor graph (NNG-SANN), which is designed to ensure the security of approximate k-nearest neighbor queries for vector data commonly used in LLMs. Specifically, a secure indexing structure and subset partitioning method are proposed based on LSH and NNG. The approach utilizes neighborhood information stored in the NNG to supplement subset data, significantly reducing the impact of false positive points generated by LSH on query results, thereby effectively improving query accuracy. To ensure data privacy, we incorporate a symmetric encryption algorithm that encrypts the data subsets obtained through greedy partitioning before storing them on the server, providing robust security guarantees. Furthermore, we construct a secure index table that enables complete candidate set retrieval through a single query, ensuring our solution completes the search process in one interaction while minimizing communication costs. Comprehensive experiments conducted on two datasets of different scales demonstrate that our proposed method outperforms existing state-of-the-art algorithms in terms of both query accuracy and security, effectively meeting the precision and security requirements for nearest neighbor queries in LLMs. Full article

A coalition in a graph G consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a dominating set but whose union $V_1\cup V_2$ is a dominating set. A coalition partition in a graph G is a vertex partition $\pi =\{V_1,V_2,\dots ,V_k\}$ such that every set $V_i\in \pi$ is not a dominating set but forms a coalition with another set $V_j\in \pi$ which is not a dominating set. The coalition number $C\left(G\right)$ equals the maximum k of a coalition partition of G. In this paper, we study the coalition number of the dth power of the n-cycle $C_n^d$, where $n\ge 3$ and $d\ge 2$. We show that $C\left(C_n^d\right)=d^2+3d+2$ for $n=2d^2+4d+2$ or $n\ge 2d^2+5d+3$, and also provide some bounds of $C\left(C_n^d\right)$ for the other cases. As a special case, we obtain the exact value of the coalition number of $C_n^2$. Full article

The Maximum Colored Cut problem aims to seek a bipartition of the vertex set of a graph, maximizing the number of colors in the crossing edges. It is a classical Max-Cut problem if the host graph is rainbow. Let $mcc\left(G\right)$ denote the maximum number of colors in a cut of an edge-colored graph G. Let $C_k$ be a cycle of length k; we say G is PC-$C_k$-free if G contains no properly colored $C_k$. We say G is a p-edge-colored graph if there exist p colors in G. In this paper, we first show that if G is a PC-$C_3$-free p-edge-colored complete 4-partite graph, then $mcc\left(G\right)=p$. Let $k\ge 3$ be an integer. Then, we show that if G is a PC-$C_4$-free p-edge-colored complete k-partite graph, then $mcc\left(G\right)\ge min\{p-1,15p/16\}$. Finally, for a p-edge-colored complete graph G, we prove that $mcc\left(G\right)\ge p-1$ if G is PC-$C_4$-free, and $mcc\left(G\right)\ge min\{p-6,7p/8\}$ if G is PC-$C_5$-free and $p\ge 7$. Full article

The exponential growth of multi-modal behavioral data in campus networks poses significant challenges for clustering analysis, including high dimensionality, redundancy, and attribute heterogeneity, which lead to degraded accuracy in existing methods. To address these issues, this study proposes a graph-theoretic subspace deep clustering framework that synergizes a deep sparse auto-encoder (DSAE) with a method of graph partitioning based on normalized cut. First, a four-layer DSAE is designed to extract discriminative features while enforcing sparsity constraints, effectively reducing data dimensionality and mitigating noise. Second, the refined subspace representations are transformed into a similarity graph, where normalized cut optimization partitions users into coherent behavioral clusters by balancing intra-cluster cohesion and inter-cluster separation. Experimental validation on three datasets—USER_DATA, MNIST, and COIL20—demonstrates the superiority of GTSDC. It achieves 91% accuracy on USER_DATA, outperforming traditional algorithms (e.g., CLIQUE, K-means) by 120% and advanced methods (e.g., deep subspace clustering) by 15%. The proposed framework not only enhances network resource allocation through behavior-aware analytics but also lays the groundwork for personalized educational services. This work bridges the gap between graph theory and deep learning, offering a scalable solution for high-dimensional behavioral pattern recognition. In simple terms, this new algorithm can more accurately analyze user behavior in campus networks. It helps universities better allocate network resources, such as ensuring smooth online classes, and can also provide personalized educational services to students according to their behavior patterns. Full article

Community detection in complex networks remains a significant challenge due to noise, outliers, and the dependency on predefined clustering parameters. This study introduces GraphDBSCAN, an adaptive community detection framework that integrates an optimized density-based clustering method with an enhanced graph partitioning approach. The proposed method refines clustering accuracy through three key innovations: (1) a K-nearest neighbor (KNN)-based strategy for automatic parameter tuning in density-based clustering, eliminating the need for manual selection; (2) a proximity-based feature extraction technique that enhances node representations while preserving network topology; and (3) an improved edge removal strategy in graph partitioning, incorporating additional centrality measures to refine community structures. GraphDBSCAN is evaluated on real-world and synthetic datasets, demonstrating improvements in modularity, noise reduction, and clustering robustness. Compared to existing methods, GraphDBSCAN consistently enhances structural coherence, reduces sensitivity to outliers, and improves community separation without requiring fixed parameter assumptions. The proposed method offers a scalable, data-driven approach to community detection, making it suitable for large-scale and heterogeneous networks. Full article

In this article, some properties of the zero-divisor graph $\Gamma \left(Z_n\right)$ are investigated when n is a square-free positive integer. It is shown that the zero-divisor graph $\Gamma \left(Z_n\right)$ of ring $Z_n$ is a $(2^k-2)$-partite graph when the prime decomposition of n contains k distinct square-free primes using the method of congruence relation. We present some examples, accompanied by graphic representations, to achieve the desired results. It is also obtained that the zero-divisor graph $\Gamma \left(Z_n\right)$ is Eulerian if n is a square-free odd integer. Since $Z_n$ is a semisimple ring when n is square-free, the results can be generalized to characterize semisimple rings and modules, as well as rings satisfying Artinian and Noetherian conditions through the properties of their zero-divisor graphs. We endeavored to show that $\Gamma \left(R\right)$ is a partite graph with a certain condition on n and also that $\Gamma \left(R\right)$ is a complete graph when $n=p^2$ for a prime p as part of a corollary. To prove these results, we employed the assistance of several theoretic congruence relations that grabbed our attention, making the investigation more interesting. Full article

Given a graph G = (V, E), the edge set can be partitioned into k dimensions, for a positive integer k. The set of all i-dimensional edges of G is a subset of E(G) denoted by E_i. A Hamiltonian cycle C on G contains all vertices on G. Let E_i(C) = E(C) ∩ E_i. For any 1 ≤ i ≤ k, C is called a dimension-balanced Hamiltonian cycle (DBH, for short) on G if ||E_i(C)| − |E_j(C)|| ≤ 1 for all 1 ≤ i < j ≤ k. The dimension-balanced cycle problem is generated with the 3-D scanning problem. Graph G is called p-node q-edge dimension-balanced Hamiltonian (p-node q-edge DBH) if it has a DBH after removing any p nodes and any q edges. G is called h-fault dimension-balanced Hamiltonian (h-fault DBH, for short) if it remains Hamiltonian after removing any h node and/or edges. The design for the network-on-chip (NoC) problem is important. One of the most famous NoC is the toroidal mesh graph T_m,n. The DBC problem on toroidal mesh graph T_m,n is appropriate for designing simple algorithms with low communication costs and avoiding congestion. Recently, the problem of a one-fault DBH on T_m,n has been studied. This paper solves the one-node one-edge DBH problem in the two-fault DBH problem on T_m,n. Full article

Our goal is to prove the Euler–Maclaurin summation formula using only the Taylor formula. Furthermore, the Euler–Maclaurin summation formula will be considered for the case of functions of the class $C^{2k}\left([a,b]\right)$. A stronger version of the estimation of the rest for functions of class $C^n\left([a,b]\right)$ is given. We will also present the application of the obtained Euler–Maclaurin summation formulas to determine the asymptotics of Riemann sums for uniform partitions of intervals of integration. This paper also introduces the concepts of the asymptotic smoothing of the graph of a given function, as well as the asymptotic uniform distribution of the positive and negative values of the given function (generalizing the concept of the symmetric distribution of these values with respect to the x-axis, as in the case of the Peano–Jordan measure). Full article

Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge set of G is partitioned into k dimensions, the problem of dimension-balanced Hamiltonian cycles arises, where the Hamiltonian cycle uses approximately the same number of edges from each dimension (differing by at most one). This paper studies whether a dimension-balanced Hamiltonian cycle (DBH) exists in toroidal mesh graphs T_m,n when a single vertex or edge is faulty, called the one-fault DBH problem. We establish that T_m,n is one-fault DBH, except in the following cases: (1) both m and n are even; (2) one of m and n is 3, while the other satisfies mod 4 = 3 and is greater than 6; (3) one of m and n is odd, while the other satisfies mod 4 = 2. Additionally, this paper resolves a conjecture from prior literature, thereby providing a complete solution to the DBP problem on T_m,n. Full article

Background: The enhancer–promoter interaction (EPI) is a critical component of gene regulatory networks, playing a significant role in understanding the complexity of gene expression. Traditional EPI prediction methods focus on one-to-one interactions, neglecting more complex one-to-many and many-to-many patterns. To address this gap, we utilize graph neural networks to comprehensively explore all interaction patterns between enhancers and promoters, capturing complex regulatory relationships for more accurate predictions. Methods: In this study, we introduce a novel EPI prediction framework, GATv2EPI, based on dynamic graph attention neural networks. GATv2EPI leverages epigenetic information from enhancers, promoters, and their surrounding regions and organizes interactions into a network to comprehensively explore complex EPI regulatory patterns, including one-to-one, one-to-many, and many-to-many relationships. To avoid overfitting and ensure diverse data representation, we implemented a connectivity-based sampling method for dataset partitioning, which constructs graphs for each chromosome and assigns entire connected subgraphs to training or test sets, thereby preventing information leakage and ensuring comprehensive chromosomal representation. Results: In experiments conducted on four cell lines—NHEK, IMR90, HMEC, and K562—GATv2EPI demonstrated superior EPI recognition accuracy compared to existing similar methods, with a training time improvement of 95.29% over TransEPI. Conclusions: GATv2EPI enhances EPI prediction accuracy by capturing complex topological structure information from gene regulatory networks through graph neural networks. Additionally, our results emphasize the importance of epigenetic features surrounding enhancers and promoters in EPI prediction. Full article

In a fuzzy graph $\mathbb{G}$, a fuzzy coalition is formed by two disjoint vertex sets ${\mathbb{V}}_{\mathtt{1}}$ and ${\mathbb{V}}_{\mathtt{2}}$, neither of which is a strongly dominating set, but the union ${\mathbb{V}}_{\mathtt{1}}\cup {\mathbb{V}}_{\mathtt{2}}$ forms a strongly dominating set. A fuzzy coalition partition of $\mathbb{G}$ is defined as $\Pi =\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$, where each set ${\mathbb{V}}_{\mathbb{i}}$ either forms a singleton strongly dominating set or is not a strongly dominating set but forms a fuzzy coalition with another non-strongly dominating set in $\Pi$. A fuzzy graph with such a fuzzy coalition partition $\Pi$ is called a fuzzy coalition graph, denoted as $\mathbb{FG}(\mathbb{G},\Pi )$. The vertex set of the fuzzy coalition graph consists of $\{{\mathbb{V}}_{\mathtt{1}},{\mathbb{V}}_{\mathtt{2}},\cdots ,{\mathbb{V}}_{\mathbb{k}}\}$, corresponding one-to-one with the sets of $\Pi$, and the two vertices are adjacent in $\mathbb{FG}(\mathbb{G},\Pi )$ if and only if ${\mathbb{V}}_{\mathbb{i}}$ and ${\mathbb{V}}_{\mathbb{j}}$ are fuzzy coalition partners in $\Pi$. This study demonstrates how fuzzy coalition graphs can model and optimize cybersecurity collaborations across critical infrastructures in smart cities, ensuring comprehensive protection against cyber threats. This study concludes that fuzzy coalition graphs offer a robust framework for optimizing collaboration and decision-making in interconnected systems like smart cities. Full article

The K-means algorithm and its variants are well-known clustering techniques. In actuarial applications, these partitioning methods can identify clusters of policies with similar attributes. The resulting partitions provide an actuarial framework for creating maps of dominant risks and unsupervised pricing grids. This research article aims to adapt well-established clustering methods to complex insurance datasets containing both categorical and numerical variables. To achieve this, we propose a novel approach based on Burt distance. We begin by reviewing the K-means algorithm to establish the foundation for our Burt distance-based framework. Next, we extend the scope of application of the mini-batch and fuzzy K-means variants to heterogeneous insurance data. Additionally, we adapt spectral clustering, a technique based on graph theory that accommodates non-convex cluster shapes. To mitigate the computational complexity associated with spectral clustering’s $O\left(n^3\right)$ runtime, we introduce a data reduction method for large-scale datasets using our Burt distance-based approach. Full article

SimRank is a widely used metric for evaluating vertex similarity based on graph topology, with diverse applications such as large-scale graph mining and natural language processing. The objective of the single-source and top-k SimRank query problem is to retrieve the kvertices with the largest SimRank to the source vertex. However, existing algorithms suffer from inefficiency as they require computing SimRank for all vertices to retrieve the top-k results. To address this issue, we propose an algorithm named HitSimthat utilizes a branch and bound strategy for the single-source and top-k query. HitSim initially partitions vertices into distinct sets based on their shortest-meeting lengths to the source vertex. Subsequently, it computes an upper bound of SimRank for each set. If the upper bound of a set is no larger than the minimum value of the current top-k results, HitSim efficiently batch-prunes the unpromising vertices within the set. However, in scenarios where the graph becomes dense, certain sets with large upper bounds may contain numerous vertices with small SimRank, leading to redundant overhead when processing these vertices. To address this issue, we propose an optimized algorithm named HitSim-OPT that computes the upper bound of SimRank for each vertex instead of each set, resulting in a fine-grained and efficient pruning process. The experimental results conducted on six real-world datasets demonstrate the performance of our algorithms in efficiently addressing the single-source and top-k query problem. Full article

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set $V\left(G\right)$ in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least $\left(n\right(k-1)+r(k-r\left)\right)/2$, where $r\equiv n(modk)$ and $0\le r\le k-1$, and the bound is tight. Then, we obtain that if G is k-colorable ($k\ge 2$) and the minimum degree of G is at least $\frac{k-2}{k-1}n$, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where $k\ge 3$. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for $k\in \{5,6\}$. Full article

In the clutter environment, the measurement of a set of multiple extended targets, with an unknown number of targets, poses challenges in partitioning, and the computational cost is high. In particular, the multiple extended target tracking method, based on distance partition, has obvious potential estimation errors when the extended targets intersect. This paper proposes a partition algorithm, based on spatio-temporal correlation, which considers the correlation between adjacent moments of the extended target and uses this prior information to divide the measurement set into a survival target measurement set and a born target measurement set for the first time. Then, the survival target measurement set is clustered by the K-means++ algorithm, and the extended target tracking is transformed into point target tracking. The born target measurement undergoes preprocessing by the DBSCAN clustering algorithm, and then uses the directed graph with shared nearest neighbors (SNN) dividing the measurement set. The method proposed in this paper significantly reduces the number of partitions and the computational time. The effectiveness of the algorithm is demonstrated through experimental simulations. Full article