Bipartite Graph Research Articles

Graph Representation Learning Enhanced Semi-Supervised Feature Selection

Feature selection is a key step in machine learning by eliminating features that are not related to the modeling target to create reliable and interpretable models. By exploring the potential complex correlations among features of unlabeled data, recently introduced self-supervision-enhanced feature selection greatly reduces the reliance on the labeled samples. However, they are generally based on the autoencoder with sample-wise self-supervision, which can hardly exploit the relations among samples. To address this limitation, this article proposes graph representation learning enhanced semi-supervised feature selection (G-FS) which performs feature selection based on the discovery and exploitation of the non-Euclidean relations among features and samples by translating unlabeled “plain” tabular data into a bipartite graph. A self-supervised edge prediction task is designed to distill rich information on the graph into low-dimensional embeddings, which remove redundant features and noise. Guided by the condensed graph representation, we propose a batch attention feature weight generation mechanism that generates more robust weights according to batch-based selection patterns rather than individual samples. The results show that G-FS achieves significant performance edges in 14 datasets compared to twelve state-of-the-art baselines, including two recent self-supervised baselines. The source code is public available at https://github.com/Icannotnamemyselff/G-FS_Graph_enhacned_feature_selection .

One-to-one Disjoint-path Covers of Leaf-sort Graphs

Parallel paths of interconnection networks have attracted much attention in parallel computing systems, they are studied in terms of disjoint paths in an undirected graph [Formula: see text]. A [Formula: see text]-disjoint path cover (k-DPC for short) of a graph [Formula: see text] is a set of [Formula: see text] (internally) disjoint paths that altogether cover every vertice of the graph. A bipartite graph [Formula: see text] is one-to-one bi-k-DPC if there is a k-DPC between any pair of vertices in different partite sets. A bipartite graph is hamiltonian laceable if there exists a hamiltonian path between any pair of vertices in different partite sets. The [Formula: see text]-dimensional leaf-sort graph [Formula: see text] has many good properties, including bipartite, symmetry, vertex-transitive. In this paper, we prove that [Formula: see text] has one-to-one bi-[Formula: see text]-DPC between any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] in different partite sets, and [Formula: see text] is hamiltonian laceable.

Metrics of positive Ricci curvature on simply‐connected manifolds of dimension 6k$6k$

AbstractA consequence of the surgery theorem of Gromov and Lawson is that every closed, simply‐connected 6‐manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature, it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article, we introduce a new description of certain ‐dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way, we obtain many new examples, both spin and nonspin, of ‐dimensional manifolds with a metric of positive Ricci curvature.

Optimal Algorithms for Online b-Matching with Variable Vertex Capacities

AbstractWe study the b-matching problem, which generalizes classical online matching introduced by Karp, Vazirani and Vazirani (STOC 1990). Consider a bipartite graph $$G=(S\dot{\cup }R,E)$$ G = ( S ∪ ˙ R , E ) . Every vertex $$s\in S$$ s ∈ S is a server with a capacity $$b_s$$ b s , indicating the number of possible matching partners. The vertices $$r\in R$$ r ∈ R are requests that arrive online and must be matched immediately to an eligible server. The goal is to maximize the cardinality of the constructed matching. In contrast to earlier work, we study the general setting where servers may have arbitrary, individual capacities. We prove that the most natural and simple online algorithms achieve optimal competitive ratios. As for deterministic algorithms, we give a greedy algorithm RelativeBalance and analyze it by extending the primal-dual framework of Devanur, Jain and Kleinberg (SODA 2013). In the area of randomized algorithms we study the celebrated Ranking algorithm by Karp, Vazirani and Vazirani. We prove that the original Ranking strategy, simply picking a random permutation of the servers, achieves an optimal competitiveness of $$1-1/e$$ 1 - 1 / e , independently of the server capacities. Hence it is not necessary to resort to a reduction, replacing every server s by $$b_s$$ b s vertices of unit capacity and to then run Ranking on this graph with $$\sum _{s\in S} b_s$$ ∑ s ∈ S b s vertices on the left-hand side. Additionally, we extend this result to the vertex-weighted b-matching problem. Technically, we formulate a new configuration LP for the b-matching problem and conduct a primal-dual analysis.

Geography, Kotzig's Nim, and variants

Geography is a combinatorial game in which two players take turns moving a token along edges of a directed graph and deleting the vertex they came from. We expand upon work by Fox and Geissler, who classified the computational complexity of determining the winner of various Geography variants given a graph. In particular, we show NP-hardness for undirected partizan Geography with free deletion on bipartite graphs and directed partizan Geography with free deletion on acyclic graphs. In addition, we study Kotzig's Nim, a special case of Geography where the vertices are labeled and moves correspond to additions by fixed amounts. We partially resolve a conjecture by Tan and Ward about games with a certain move set.

Edge Manipulations for the Maximum Vertex-Weighted Bipartite \(b\) -matching

In this paper, we explore the Mechanism Design aspects of the Maximum Vertex-weighted \(b\) -Matching (MVbM) problem on bipartite graphs \((A\cup T,E)\) . The set \(A\) comprises agents, while \(T\) represents tasks. The set \(E\) , which connects \(A\) and \(T\) , is the private information of either agents or tasks. In this framework, we investigate three mechanisms – \(\mathbb{M}_{BFS}\) , \(\mathbb{M}_{DFS}\) , and \(\mathbb{M}_{G}\) . We examine scenarios in which either agents or tasks are strategic and report their adjacent edges to one of the three mechanisms. In both cases, we assume that the strategic entities are bounded by their statements: they can hide edges, but they cannot report edges that do not exist. First, we consider the case in which agents can manipulate. In this framework, \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) are optimal but not truthful. By characterizing the Nash Equilibria induced by \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) , we reveal that both mechanisms have a Price of Anarchy ( \(PoA\) ) and Price of Stability ( \(PoS\) ) of \(2\) . These efficiency guarantees are tight; no deterministic mechanism can achieve a lower \(PoA\) or \(PoS\) . In contrast, the third mechanism, \(\mathbb{M}_{G}\) , is not optimal, but truthful and its approximation ratio is \(2\) . We demonstrate that this ratio is optimal; no deterministic and truthful mechanism can outperform it. We then shift our focus to scenarios where tasks can exhibit strategic behaviour. In this case, \(\mathbb{M}_{BFS}\) , \(\mathbb{M}_{DFS}\) , and \(\mathbb{M}_{G}\) all maintain truthfulness, making \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) truthful and optimal mechanisms. In conclusion, we investigate the manipulability of \(\mathbb{M}_{BFS}\) and \(\mathbb{M}_{DFS}\) through experiments on randomly generated graphs. We observe that (i) \(\mathbb{M}_{BFS}\) is less prone to be manipulated by the first agent than \(\mathbb{M}_{DFS}\) , and (ii) \(\mathbb{M}_{BFS}\) is more manipulable on instances in which the total capacity of the agents is equal to the number of tasks. 1

Unraveling Incommensurate Spatial Partitions: A Bipartite Graph Approach to School-Neighborhood Interactions and Their Impacts

Unraveling Incommensurate Spatial Partitions: A Bipartite Graph Approach to School-Neighborhood Interactions and Their Impacts

A Structured Bipartite Graph Learning method for ensemble clustering

Given a set of base clustering results, conventional bipartite graph-based ensemble clustering methods typically require computing a sample-cluster similarity matrix from each base clustering result. These matrices are then either concatenated or averaged to form a bipartite weight matrix, which is used to create a bipartite graph. Graph-based partition techniques are subsequently applied to this graph to obtain the final clustering result. However, these methods often suffer from unreliable base clustering results, making it challenging to identify a clear cluster structure due to the variations in cluster structures across the base results. In this paper, we propose a novel Structured Bipartite Graph Learning (SBGL) method. Our approach begins by computing a sample-cluster similarity matrix from each base clustering result and constructing a base bipartite graph from each of these matrices. We assume these base bipartite graphs contain a set of latent clusters and project them into a set of sample-latent-cluster bipartite graphs. These new graphs are then ensembled into a bipartite graph with a distinct cluster structure, from which the final set of clusters is derived. Our method allows for different numbers of clusters across base clusterings, leading to improved performance. Experimental results on both synthetic and real-world datasets demonstrate the superior performance of our new method.

Semi-supervised multi-view feature selection with adaptive similarity fusion and learning

Existing multi-view semi-supervised feature selection methods typically need to calculate the inversion of high-order dense matrices, rendering them impractical for large-scale applications. Meanwhile, traditional works construct similarity graphs on different views and directly fuse these graphs from the view level, ignoring the differences among samples in various views and the interplay between graph learning and feature selection. Consequently, both the reliability of graphs and the discrimination of selected features are compromised. To address these issues, we propose a novel multi-view semi-supervised feature selection with Adaptive Similarity Fusion and Learning (ASFL) for large-scale tasks. Specifically, ASFL constructs bipartite graphs for each view and then leverages the relationships between samples and anchors to align anchors and graphs across different views, preserving the complementarity and consistency among views. Moreover, an effective view-to-sample fusion manner is designed to coalesce the aligned graphs while simultaneously exploiting the neighbor structures in projection subspaces to construct the joint graph compatible across views, reducing the adverse effects of noisy features and outliers. By incorporating bipartite graph fusion and learning, label propagation, and l2,0-norm multi-view feature selection into a unified framework, ASFL not only avoids the expensive computation in the solution procedures but also enhances the quality of selected features. An effective optimization strategy with fast convergence is developed to solve the objective function, and experimental results validate its efficiency and effectiveness over state-of-the-art methods.

On comparison between the distance energies of a connected graph

Let G be a simple connected graph of order n having Wiener index W(G). The distance, distance Laplacian and the distance signless Laplacian energies of G are respectively defined asDE(G)=∑i=1n|υiD|,DLE(G)=∑i=1n|υiL−Tr‾|andDSLE(G)=∑i=1n|υiQ−Tr‾|, where υiD,υiL and υiQ,1≤i≤n are respectively the distance, distance Laplacian and the distance signless Laplacian eigenvalues of G and Tr‾=2W(G)n is the average transmission degree. In this paper, we will study the relation between DE(G), DLE(G) and DSLE(G). We obtain some necessary conditions for the inequalities DLE(G)≥DSLE(G),DLE(G)≤DSLE(G),DLE(G)≥DE(G) and DSLE(G)≥DE(G) to hold. We will show for graphs with one positive distance eigenvalue the inequality DSLE(G)≥DE(G) always holds. Further, we will show for the complete bipartite graphs the inequality DLE(G)≥DSLE(G)≥DE(G) holds. We end this paper by computational results on graphs of order at most 6.

On the $Q$-Polynomial Property of the Full Bipartite Graph of a Hamming Graph

The $Q$-polynomial property is an algebraic property of distance-regular graphs, that was introduced by Delsarte in his study of coding theory. Many distance-regular graphs admit the $Q$-polynomial property. Only recently the $Q$-polynomial property has been generalized to graphs that are not necessarily distance-regular. In [J. Combin. Theory Series. A, 205:105872, 2024] it was shown that the graphs arising from the Hasse diagrams of the so-called attenuated space posets are $Q$-polynomial. These posets could be viewed as $q$-analogs of the Hamming posets, which were not studied in that paper. The main goal of this article is to fill this gap by showing that the graphs arising from the Hasse diagrams of the Hamming posets are $Q$-polynomial.

Attribute enhanced random walk for community detection in attributed networks

Traditional community detection methods often rely solely on network topology, potentially overlooking the influence of attributes. However, community detection in attributed networks presents a unique challenge due to the complex interplay between network topology and node attributes. In this paper, we present AERW, a novel approach that integrates both network topology and node attributes for effective community detection. AERW leverages the principle of homophily by aggregating neighbor attributes within triangles to capture higher-order structural information. Then constructing a bipartite graph to effectively integrate topological and attribute information, ensuring that both aspects are equally represented in the community detection process. Next AERW utilizes hierarchical clustering applied to the probability transition matrix derived from the random walk, enabling the identification of community structures without needing prior knowledge of the number of communities. Extensive experimentation across synthetic and real-world networks underscores AERW’s efficacy, establishing it as a superior approach in community detection within attributed networks.

Edge computing task scheduling method based on user’s social relations: a construction and solution for Smart City Library

In the realm of the development of a Smart City Library, the integration of robust edge computing is vital. The research suggests a novel task-scheduling model for edge computing, leveraging user’s social relationships. Analyzing these connections involves constructing a user’s social relationship graph by implementing mathematical convolution and the Jaccard similarity ratio. This precise quantification of social ties ensures secure and reliable task scheduling. An equipment connection graph of a user equipment service is also crafted based on Euclidean distance, aligning task scheduling with device-to-device (D2D) communication conditions. Combining a user’s social relationship graph and a user’s device-service device connection graph creates a task-device bipartite graph. On the other hand, the calculation of a task execution cost and edge weight determination finalize a scheduling model. Implementing the proposed method for constructing a model for edge computing task scheduling based on utilizing the Kuhn–Munkres (KM) algorithm demonstrates positive impacts, which are few delays and less energy consumption, on edge computing task scheduling. For instance, when the social threshold score changes from 02. To 0.6, the total task execution delay time increases from 23 to 32, which is the best when compared with other algorithms. The approach strengthens security and reliability while decreasing task execution delays and energy consumption. This research advances edge computing for Smart City Libraries, promising transformative implications.

Orthogonal Decompositions of Regular Graphs and Designing Tree-Hamming Codes

The paper introduces a concept of graph decomposition. That is orthogonal decompositions. Orthogonal decompositions of a graph H is a partitioning H into subgraphs of H such that any two subgraphs intersect in at most one edge. This decompositions are called G-orthogonal decompositions of H if and only if every subgraph in such decompositions is isomorphic to the graph G. Such decomposition appear in a lot of applications; statistics, information theory, in the theory of experimental design and many others. An approach of constructing orthogonal decompositions of regular graph is introduced here. Application of this approach for constructing tree -- orthogonal decompositions of complete bipartite graph is considered. Further, the use of orthogonal decompositions for designing hamming tree-codes is also discussed along with examples. The study shows that such codes have an efficient properties when are used to detect and correct the errors that may occur during a transmission of data through a network. Furthermore, we present a method for the recursive construction of orthogonal decompositions.

Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.

Zagreb Polynomials, Co-Polynomials of Bipartite and Strongly Bipartite Graphs

First Zagreb polynomial of a graph G with vertex set V(G) and edge set E(G) is defined as M1(G,x) =∑uv∈E(G)xdu+dv and the first Zagreb index can be obtained from it as M1(G) = M1(G,x)|x=1. In this paper Zagreb polynomials, co-polynomials and corresponding Zagreb indices, co-indices are obtained for path, cycle, double and strong double graphs.

Multimodal Data Mining Based on Self Attention Feature Alignment

This study explores the application of multimodal data mining in text and image vector processing, aiming to improve the depth and breadth of data analysis by integrating information from different data types. We use the FairFace dataset combined with the CLIP model encoding layer to obtain text and image vectors, and use the K-Means clustering algorithm to achieve vector dimensionality reduction. Subsequently, we introduced the bipartite graph matching algorithm to achieve maximum matching between text vectors and image vectors, and calculated the contrastive learning loss and similarity loss. The entire process covers steps such as data preparation, feature extraction, vector dimensionality reduction, matching algorithms, and loss assessment, constructing a complete text and image matching task process. Our research contributions include using K-Means clustering algorithm to achieve vector dimensionality reduction, as well as introducing bipartite graph matching algorithm and calculating two types of losses in text and image vector matching, further improving matching quality.

Interpretable Dynamic Directed Graph Convolutional Network for Multi-Relational Prediction of Missense Mutation and Drug Response.

Tumor heterogeneity presents a significant challenge in predicting drug responses, especially as missense mutations within the same gene can lead to varied outcomes such as drug resistance, enhanced sensitivity, or therapeutic ineffectiveness. These complex relationships highlight the need for advanced analytical approaches in oncology. Due to their powerful ability to handle heterogeneous data, graph convolutional networks (GCNs) represent a promising approach for predicting drug responses. However, simple bipartite graphs cannot accurately capture the complex relationships involved in missense mutation and drug response. Furthermore, Deep learning models for drug response are often considered "black boxes", and their interpretability remains a widely discussed issue. To address these challenges, we propose an Interpretable Dynamic Directed Graph Convolutional Network (IDDGCN) framework, which incorporates four key features: (1) the use of directed graphs to differentiate between sensitivity and resistance relationships, (2) the dynamic updating of node weights based on node-specific interactions, (3) the exploration of associations between different mutations within the same gene and drug response, and (4) the enhancement of interpretability models through the integration of a weighted mechanism that accounts for the biological significance, alongside a ground truth construction method to evaluate prediction transparency. The experimental results demonstrate that IDDGCN outperforms existing state-of-the-art models, exhibiting excellent predictive power. Both qualitative and quantitative evaluations of its interpretability further highlight its ability to explain predictions, offering a fresh perspective for precision oncology and targeted drug development.

A distance-based network activity correlation framework for defeating anonymization overlays

As the effectiveness of modern Internet-based anonymization infrastructures grows, law enforcement agencies are experiencing a progressive erosion of their surveillance capabilities. This can severely undermine their efforts to prevent and investigate various types of unlawful activities, potentially increasing the impunity of organized criminal networks. Balancing the legitimate privacy needs of individuals with the imperative to maintain public safety and combat criminal behavior in the digital world remains a complex tradeoff for both policymakers and technologists who need to find a systematic and reliable way to link the traffic traces associated with criminal activities to their anonymized origins. Accordingly, this paper presents a simple but very effective de-anonymization approach capable of associating traffic traces captured at the edge of the overlay infrastructures, in correspondence with the true origins, to those captured in correspondence with the destinations. The approach is based on determining the minimum-distance pairs within a complete bipartite graph in which the traffic traces are the nodes. Experiments with different distance functions, applied in varied ways, show that the resulting framework appears to be a promising solution that is scalable and easily deployable on real-life network equipment.

List coloring based algorithm for the Futoshiki puzzle

Given a graph G=(V, E) and a list of available colors L(v) for each vertex v\in V, where L(v) \subseteq {1, 2, ..., k}, List k-Coloring refers to the problem of assigning colors to the vertices of $G$ so that each vertex receives a color from its own list and no two neighboring vertices receive the same color. The decision version of the problem, List k-Coloring, is NP-complete even for bipartite graphs. As an application of list coloring problem we are interested in the Futoshiki Problem. Futoshiki is an NP-complete Latin Square Completion Type Puzzle. Considering Futoshiki puzzle as a constraint satisfaction problem, we first give a list coloring based algorithm for it which is efficient for small boards of fixed size. To thoroughly investigate the efficiency of our algorithm in comparison with a proposed backtracking-based algorithm, we conducted a substantial number of computational experiments at different difficulty levels, considering varying numbers of inequality constraints and given values. Our results from the extensive range of experiments indicate that the list coloring-based algorithm is much more efficient.