Graph Partitioning Research Articles

Partitioning problems via random processes

AbstractThere are a number of well‐known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3‐colouring such that for every vertex , at most half of the out‐neighbours of have the same colour as . As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every , all but finitely many ‐regular graphs have a partition into two non‐empty parts such that for every vertex , at least half of the neighbours of lie in the same part as . We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for Erdős–Rényi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of ‘exceptional vertices’. Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a personality‐changing scheme: we ‘forget’ certain information based on the state of a Markov chain, giving us more independence to work with.

Bad Local Minima Exist in the Stochastic Block Model

AbstractWe study the disassortative stochastic block model with three communities, a well-studied model of graph partitioning and Bayesian inference for which detailed predictions based on the cavity method exist (Decelle et al. in Phys Rev E 84:066106, 2011). We provide strong evidence that for a part of the phase where efficient algorithms exist that approximately reconstruct the communities, inference based on maximum a posteriori (MAP) fails. In other words, we show that there exist modes of the posterior distribution that have a vanishing agreement with the ground truth. The proof is based on the analysis of a graph colouring algorithm from Achlioptas and Moore (J Comput Syst Sci 67:441–471, 2003).

Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

ABSTRACTIn this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.

On perfectly friendly bisections of random graphs

On perfectly friendly bisections of random graphs

Maximum Bisections of Graphs with Girth at Least Six

Maximum Bisections of Graphs with Girth at Least Six

A Proactive Decoy Selection Scheme for Cyber Deception using MITRE ATT&CK

Cyber deception allows compensating the late response of defenders countermeasures to the ever evolving tactics, techniques, and procedures (TTPs) of attackers. This proactive defense strategy employs decoys resembling legitimate system components to lure stealthy attackers within the defender environment, slowing and/or denying the accomplishment of their goals. In this regard, the selection of decoys that can expose the techniques used by malicious users plays a central role to incentivize their engagement. However, this is a difficult task to achieve in practice, since it requires an accurate and realistic modeling of the attacker capabilities and his possible targets. In this work, we tackle this challenge and we design a decoy selection scheme that is supported by an adversarial modeling based on empirical observation of real-world attackers. We take advantage of a domain-specific threat modeling language using MITRE ATT&CK© framework as source of attacker TTPs targeting enterprise systems. In detail, we extract the information about the execution preconditions of each technique as well as its possible effects on the environment to generate attack graphs modeling the adversary capabilities. Based on this, we formulate a graph partition problem that minimizes the number of decoys detecting a corresponding number of techniques employed in various attack paths directed to specific targets. We compare our optimization-based decoy selection approach against several benchmark schemes that ignore the preconditions between the various attack steps. Results reveal that the proposed scheme provides the highest interception rate of attack paths using the lowest amount of decoys.

Large scale multi-GPU based parallel traffic simulation for accelerated traffic assignment and propagation

Traffic simulation is a critical tool for congestion analysis, travel time estimation, and route optimization in urban planning, benefiting navigation apps, transportation network companies, and state agencies. Traditionally, traffic micro-simulation frameworks are based on road segments and can only support a limited number of main roads. Efficient traffic simulation on a regional scale remains a significant challenge due to the complexity of urban mobility and the large scale of spatiotemporal data. This paper introduces a Large Scale Multi-GPU Parallel Computing based Regional Scale Traffic Simulation Framework (LPSim), which leverages graphical processing unit (GPU) parallel computing to address these challenges. LPSim utilizes a multi-GPU architecture to simulate extensive and dynamic traffic networks with high fidelity and reduced computation time. Using the parallel processing capabilities of GPUs, LPSim can perform tens of millions of individual vehicle dynamics simulations simultaneously, significantly outperforming traditional CPU-based approaches. The framework is designed to be scalable and can easily accommodate the increasing complexity of traffic simulations. We present the theory behind GPU-based traffic simulation, the architecture of single- and multi-GPU based simulations, and the graph partition strategies that enhance computation resource load balance. Our experimental results demonstrate the effectiveness of LPSim in simulating large-scale traffic scenarios. LPSim is capable of completing simulations of 2.82 million trips in just 6.28 min on a single GPU machine equipped with 5120 CUDA cores (Tesla V100-SXM2). Furthermore, utilizing a Google Cloud instance with two NVIDIA V100 GPUs, which collectively offer 10240 CUDA cores, LPSim successfully simulates 9.01 million trips within 21.16 min. We further tested our simulator with the same demand on dual NVIDIA A100-PCIE-40GB GPUs, which finished the simulation in 0.0398 h, approximately 113 times faster than the same simulation scenario running on an Intel(R) Xeon(R) Gold 6326 CPU @ 2.90 GHz, which takes 4.49 h to complete. This performance not only demonstrates its speed and scalability advantages over traditional simulation techniques but also highlights LPSim’s unique position as the first traffic simulation framework that is scalable for both single- and multiple-GPU configurations. Consequently, LPSim provides an invaluable tool for individuals and extensive research teams alike, enabling the acquisition of large-scale traffic simulation results in a time-efficient manner. LPSim code is available at: https://github.com/Xuan-1998/LPSim

Smart-KG: Partition-Based Linked Data Fragments for querying knowledge graphs

RDF and SPARQL provide a uniform way to publish and query billions of triples in open knowledge graphs (KGs) on the Web. Yet, provisioning of a fast, reliable, and responsive live querying solution for open KGs is still hardly possible through SPARQL endpoints alone: while such endpoints provide a remarkable performance for single queries, they typically can not cope with highly concurrent query workloads by multiple clients. To mitigate this, the Linked Data Fragments (LDF) framework sparked the design of different alternative low-cost interfaces such as Triple Pattern Fragments (TPF), that partially offload the query processing workload to the client side. On the downside, such interfaces still come with the expense of unnecessarily high network load due to the necessary transfer of intermediate results to the client, leading to query performance degradation compared with endpoints. To address this problem, in the present work, we investigate alternative interfaces, refining and extending the original TPF idea, which also aims at reducing server-resource consumption, by shipping query-relevant partitions of KGs from the server to the client. To this end, first, we align formal definitions and notations of the original LDF framework to uniformly present existing LDF implements and such “partition-based” LDF approaches. These novel LDF interfaces retrieve, instead of the exact triples matching a particular query pattern, a subset of pre-materialized, compressed, partitions of the original graph, containing all answers to a query pattern, to be further evaluated on the client side. As a concrete representative of partition-based LDF, we present smart-KG+, extending and refining our prior work (In WWW ’20: The Web Conference 2020 (2020) 984–994 ACM / IW3C2) in several respects. Our proposed approach is a step forward towards a better-balanced share of the query processing load between clients and servers by shipping graph partitions driven by the structure of RDF graphs to group entities described with the same sets of properties and classes, resulting in significant data transfer reduction. Our experiments demonstrate that the smart-KG+ significantly outperforms existing Web SPARQL interfaces on both pre-existing benchmarks for highly concurrent query execution as well as an accustomed query workload inspired by query logs of existing SPARQL endpoints.

BUGTGNN: Bayesian Uncertainty and Game Theory for Enhancing Graph Neural Networks through Mathematical Analysis

The ongoing COVID-19 pandemic has emphasized the critical need for rapid and accurate diagnostic methods. Precise classification of chest X-ray images into COVID-19 and non-COVID-19 cases serves as a pivotal tool in effective disease management and control. Existing methods often suffer from trade-offs between accuracy, precision, and computational efficiency, hindering their practical utility. Current approaches mainly rely on traditional machine learning algorithms or Convolutional Neural Networks (CNNs), which while effective, still present limitations in terms of sensitivity, specificity, and computational speed. These constraints necessitate the exploration of innovative techniques for improving classification metrics across multiple dimensions. In this work, we introduce a novel framework for optimizing Graph Neural Networks (GNNs) through mathematical analysis, specifically incorporating spectral methods, dynamic graph sparsification, game-theoretic attention mechanisms, Bayesian uncertainty models, and advanced graph partitioning techniques. When applied to the classification of COVID-19 chest X-rays, our model demonstrated significant improvements—increasing precision by 8.3%, accuracy by 8.5%, recall by 4.9%, specificity by 4.5%, and the Area Under the Curve (AUC) by 5.9%, while simultaneously reducing computational delay by 10.5% across multiple datasets. The proposed optimization strategies showcase the power of interdisciplinary approaches in advancing machine learning techniques for medical applications. The demonstrated improvements in classification metrics and computational efficiency highlight the model's potential for broader adoption in healthcare settings, providing a robust, fast, and more accurate tool for COVID-19 diagnosis.

Deep spectral improvement for unsupervised image instance segmentation.

Recently, there has been growing interest in deep spectral methods for image localization and segmentation, influenced by traditional spectral segmentation approaches. These methods reframe the image decomposition process as a graph partitioning task by extracting features using self-supervised learning and utilizing the Laplacian of the affinity matrix to obtain eigensegments. However, instance segmentation has received less attention than other tasks within the context of deep spectral methods. This paper addresses that not all channels of the feature map extracted from a self-supervised backbone contain sufficient information for instance segmentation purposes. Some channels are noisy and hinder the accuracy of the task. To overcome this issue, this paper proposes two channel reduction modules, Noise Channel Reduction (NCR) and Deviation-based Channel Reduction (DCR). The NCR retains channels with lower entropy, as they are less likely to be noisy, while DCR prunes channels with low standard deviation, as they lack sufficient information for effective instance segmentation. Furthermore, the paper demonstrates that the dot product, commonly used in deep spectral methods, is not suitable for instance segmentation due to its sensitivity to feature map values, potentially leading to incorrect instance segments. A novel similarity metric called Bray-curtis over Chebyshev (BoC) is proposed to address this issue. This metric considers the distribution of features in addition to their values, providing a more robust similarity measure for instance segmentation. Quantitative and qualitative results on the Youtube-VIS 2019 and OVIS datasets highlight the improvements achieved by the proposed channel reduction methods and using BoC instead of the conventional dot product for creating the affinity matrix. These improvements regarding mean Intersection over Union (mIoU) and extracted instance segments are observed, demonstrating enhanced instance segmentation performance. The code is available on: https://github.com/farnooshar/SpecUnIIS.

Partitioning the projective plane into two incidence‐rich parts

AbstractAn internal or friendly partition of a vertex set of a graph is a partition to two nonempty sets such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.

An end-to-end bi-objective approach to deep graph partitioning

Graphs are ubiquitous in real-world applications, such as computation graphs and social networks. Partitioning large graphs into smaller, balanced partitions is often essential, with the bi-objective graph partitioning problem aiming to minimize both the “cut” across partitions and the imbalance in partition sizes. However, existing heuristic methods face scalability challenges or overlook partition balance, leading to suboptimal results. Recent deep learning approaches, while promising, typically focus only on node-level features and lack a truly end-to-end framework, resulting in limited performance. In this paper, we introduce a novel method based on graph neural networks (GNNs) that leverages multilevel graph features and addresses the problem end-to-end through a bi-objective formulation. Our approach explores node-, local-, and global-level features, and introduces a well-bounded bi-objective function that minimizes the cut while ensuring partition-wise balance across all partitions. Additionally, we propose a GNN-based deep model incorporating a Hardmax operator, allowing the model to optimize partitions in a fully end-to-end manner. Experimental results on 12 datasets across various applications and scales demonstrate that our method significantly improves both partitioning quality and scalability compared to existing bi-objective and deep graph partitioning baselines.

Efficient State Sharding in Blockchain via Density-based Graph Partitioning

Efficient State Sharding in Blockchain via Density-based Graph Partitioning

An Improved Driving Training-Based Optimization Algorithm for the Minimum Gap Graph Connected Partition Problem

In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.

Spectral Bipartition via Gap Cut on DNA Sequences

Deoxyribonucleic Acid (DNA) and graph partitioning are two distinct fields of study which can be linked in the structure of biological networks. Graph partitioning has been extensively studied but not its application in the biological field. This research explored on the application of spectral graph partitioning in DNA splicing, aiming to simulate the cleavage of DNA by performing spectral bipartition on DNA sequences in a DNA splicing system. This research incorporates Fiedler theory and algebraic graph theory, which are commonly utilized in network analysis and the analysis of graph connectivity. Some DNA sequences of even length are selected and expressed in graphical representations. The adjacency matrix, Laplacian matrix, and degree matrix are computed from the graphs, as well as the Fiedler value and Fiedler vector associated with the graphs. Gap cut is used as a method of spectral bipartition which produces two partitions of DNA sequence of unequal lengths. The generalizations of gap cut on DNA sequences of even length are provided as lemmas and theorem.

Auto-weighted Graph Reconstruction for efficient ensemble clustering

Ensemble clustering (EC) is an important field in data clustering, which aims to combine multiple base clustering results of a given dataset into a single consensus result. Previous studies have shown that one of the effective methods is to convert the base clustering results into a graph partitioning problem. Existing spectral ensemble clustering methods typically obtain a spectral embedding matrix for the graph and then perform k-means, separating the graph construction and clustering tasks into two distinct stages. To address these challenges, we propose a novel and efficient ensemble clustering method named Auto-weighted Graph Reconstruction for efficient ensemble clustering. This method integrates weighted base clusters, consensus distance graph construction, and clustering into a unified framework. A consensus distance graph is constructed, from which a stable and discrete label matrix is derived using spectral clustering. An effective optimization algorithm is then employed to address the resulting problem. Finally, the experimental results on eight real-world datasets verify the effectiveness and superiority of the proposed method.

Dynamic Partitioning of Graphs Based on Multivariate Blood Glucose Data—A Graph Neural Network Model for Diabetes Prediction

Dynamic Partitioning of Graphs Based on Multivariate Blood Glucose Data—A Graph Neural Network Model for Diabetes Prediction

Parallel finite-element codes for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present and distribute a parallel finite-element toolbox written in the free software FreeFEM for computing the Bogoliubov-de Gennes (BdG) spectrum of stationary solutions to one- and two-component Gross-Pitaevskii (GP) equations, in two or three spatial dimensions. The parallelization of the toolbox relies exclusively upon the recent interfacing of FreeFEM with the PETSc library. The latter contains itself a wide palette of state-of-the-art linear algebra libraries, graph partitioners, mesh generation and domain decomposition tools, as well as a suite of eigenvalue solvers that are embodied in the SLEPc library. Within the present toolbox, stationary states of the GP equations are computed by a Newton method. Branches of solutions are constructed using an adaptive step-size continuation algorithm. The combination of mesh adaptivity tools from FreeFEM with the parallelization features from PETSc makes the toolbox efficient and reliable for the computation of stationary states. Their BdG spectrum is computed using the SLEPc eigenvalue solver. We perform extensive tests and validate our programs by comparing the toolbox's results with known theoretical and numerical findings that have been reported in the literature. Program summaryProgram Title: FFEM_BdG_ddm_toolbox.zipCPC Library link to program files:https://doi.org/10.17632/w9hg964wpb.1Licensing provisions: GPLv3Programming language:FreeFEM (v 4.12) free software (www.freefem.org)Nature of problem: Among the plethora of configurations that may exist in Gross-Pitaevskii (GP) equations modeling one or two-component Bose-Einstein condensates, only the ones that are deemed spectrally stable (or even, in some cases, weakly unstable) have high probability to be observed in realistic ultracold atoms experiments. To investigate the spectral stability of solutions requires the numerical study of the linearization of GP equations, the latter commonly known as the Bogoliubov-de Gennes (BdG) spectral problem. The present software offers an efficient and reliable tool for the computation of eigenvalues (or modes) of the BdG problem for a given two- or three-dimensional GP configuration. Then, the spectral stability (or instability) can be inferred from its spectrum, thus predicting (or not) its observability in experiments.Solution method: The present toolbox in FreeFEM consists of the following steps. At first, the GP equations in two (2D) and three (3D) spatial dimensions are discretized by using P2 (piece-wise quadratic) Galerkin triangular (in 2D) or tetrahedral (in 3D) finite elements. For a given configuration of interest, mesh adaptivity in FreeFEM is deployed in order to reduce the size of the problem, thus reducing the toolbox's execution time. Then, stationary states of the GP equations are obtained by a Newton method whose backbone involves the use of a reliable and efficient linear solver judiciously selected from the PETSc1 library. Upon identifying stationary configurations, to trace branches of such solutions a parameter continuation method over the chemical potential in the GP equations (effectively controlling the number of atoms in a BEC) is employed with step-size adaptivity of the continuation parameter. Finally, the computation of the stability of branches of solutions (i.e. the BdG spectrum), is carried out by accurately solving, at each point in the parameter space, the underlying eigenvalue problem by using the SLEPc2 library. Three-dimensional computations are made affordable in the present toolbox by using the domain decomposition method (DDM). In the course of the computation, the toolbox stores not only the solutions but also the eigenvalues and respective eigenvectors emanating from the solution to the BdG problem. We offer examples for computing stationary configurations and their BdG spectrum in one- and two-component GP equations.Running time: From minutes to hours depending on the mesh resolution and space dimension.

Co-Clustering by Directly Solving Bipartite Spectral Graph Partitioning.

Bipartite spectral graph partitioning (BSGP) method as a co-clustering method, has been widely used in document clustering, which simultaneously clusters documents and words by making full use of the duality between documents and words. It consists of two steps: 1) graph construction and 2) singular value decomposition on the bipartite graph to compute a continuous cluster assignment matrix, followed by post-processing to get the discrete solution. However, the generated graph is unstructured and fixed. It heavily relies on the quality of the graph construction. Moreover, the two-stage process may deviate from directly solving the primal problem. In order to tackle these defects, a novel bipartite graph partitioning method is proposed to learn a bipartite graph with exactly c connected components (c is the number of clusters), which can obtain clustering results directly. Furthermore, it is experimentally and theoretically proved that the solution of the proposed model is the discrete solution of the primal BSGP problem for a special situation. Experimental results on synthetic and real-world datasets exhibit the superiority of the proposed method.

Achromatic colorings of polarity graphs

A complete partition of a graph G is a partition of the vertex set such that there is at least one edge between any two parts. The largest r such that G has a complete partition into r parts, each of which is an independent set, is the achromatic number of G. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of C4-free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.