Graph Partitioning Research Articles

Edge-Based Minimal k-Core Subgraph Search

In social networks, k-core is commonly used to measure the stability of a network. When a user in a k-core leaves the network, other users may follow the user to leave. Hence, maintaining a key user is important to keep the stability of a network. It is known that an edge between two users models the relationship between the two users. In some scenarios, maintaining a relationship comes at a cost. Therefore, selectively in maintaining the relationships between users is crucial. In this paper, we for the first time conceive the concept of an edge-based minimal k-core model. An edge-based minimal k-core is a k-core with a minimal number of edges. In other words, removing any edge in an edge-based minimal k-core would make it not be a k-core any more. Based on this model, we proposed two problems, namely, an edge-based minimal k-core subgraph search (EMK-SS) and an edge-based minimal k-core subgraph search with a query node q (EMK-q-SS). Given a graph G, an integer k, and a query node (a key user) q, the EMK-q-SS problem is to find all the edge-based minimal k-cores containing the query node q, and the EMK-SS problem is to find all the edge-based minimal k-cores. We also theoretically prove that the two problems are both NP-complete. To deal with the proposed problems, we design two novel algorithms, namely the edge deletion algorithm and edge extension algorithm. Further, a graph partitioning technique is employed to speed up the computation. Comprehensive experiments on synthetic and real networks are conducted to demonstrate the effect and efficiency of our proposed methods.

Clustered Cell-Free Networking: A Graph Partitioning Approach

By moving to millimeter wave (mmWave) frequencies, base stations (BSs) will be densely deployed to provide seamless coverage in sixth generation (6G) mobile communication systems, which, unfortunately, leads to severe cell-edge problem. In addition, with massive multiple-input-multiple-output (MIMO) antenna arrays employed at BSs, the beamspace channel is sparse for each user, and thus there is no need to serve all the users in a cell by all the beams therein jointly. Therefore, it is of paramount importance to develop a flexible clustered cell-free networking scheme that can decompose the whole network into a number of weakly interfered small subnetworks operating independently and in parallel. Given a per-user rate constraint for service quality guarantee, this paper aims to maximize the number of decomposed subnetworks so as to reduce the signaling overhead and system complexity as much as possible. By formulating it as a bipartite graph partitioning problem, a rate-constrained network decomposition (RC-NetDecomp) algorithm is proposed, which can smoothly tune the network structure from the current cellular network with simple beam allocation to a fully cooperative network by increasing the required per-user rate. Simulation results demonstrate that the proposed RC-NetDecomp algorithm outperforms existing baselines in terms of average per-user rate, fairness among users and energy efficiency.

Identifying influential nodes in complex networks based on network embedding and local structure entropy

The identification of influential nodes in complex networks remains a crucial research direction, as it paves the way for analyzing and controlling information diffusion. The currently presented network embedding algorithms are capable of representing high-dimensional and sparse networks with low-dimensional and dense vector spaces, which not only keeps the network structure but also has high accuracy. In this work, a novel centrality approach based on network embedding and local structure entropy, called the ELSEC, is proposed for capturing richer information to evaluate the importance of nodes from the view of local and global perspectives. In short, firstly, the local structure entropy is used to measure the self importance of nodes. Secondly, the network is mapped to a vector space to calculate the Manhattan distance between nodes by using the Node2vec network embedding algorithm, and the global importance of nodes is defined by combining the correlation coefficients. To reveal the effectiveness of the ELSEC, we select three types of algorithms for identifying key nodes as contrast approaches, including methods based on node centrality, optimal decycling based algorithms and graph partition based methods, and conduct experiments on ten real networks for correlation, ranking monotonicity, accuracy of high ranking nodes and the size of the giant connected component. Experimental results show that the ELSEC algorithm has excellent ability to identify influential nodes.

Connectivity probability evaluation of a large-scale highway bridge network using network decomposition

The existing NP-hard problem makes it difficult to evaluate the connectivity probability of large-scale networks, causing great barriers to evaluating and ensuring network safety. In this study, the connectivity probability for a highway bridge network composed of 1772 bridges is evaluated using network decomposition. First, the multilevel k-way graph partition is employed recursively to decompose the network into several approximately equally-sized subnetworks and minimum edge-cuts in series. Then, the decomposed network connectivity probability could be achieved in two steps: subnet and simplified network evaluations. In the subnet evaluation step, the subnet states with and without edge-cuts are judged respectively. Different from the existing connectivity binary definition as connected or disconnected, three states are redefined using adjacent matrices for each subnet: subnet with and without edge-cuts both connected (CCS), both disconnected (DDS), subnet with edge-cuts connected while without edge-cuts disconnected (DCS). For DDS would inevitably lead to the disconnection of the bridge network, such state wouldn't be enumerated in the following step for efficiency, while the CCS and DCS would be further represented by the terminal nodes. In the simplified network evaluation step, the highway bridge network connectivity can be thoroughly represented by treating terminal nodes and edge-cuts as a serial simplified network whose connectivity probability could be calculated by analyzing the limited states. The connectivity probability evaluation shows high efficiency and accuracy in the investigated bridge network.

Adaptive group testing strategy for infectious diseases using social contact graph partitions

Mass testing is essential for identifying infected individuals during an epidemic and allowing healthy individuals to return to normal social activities. However, testing capacity is often insufficient to meet global health needs, especially during newly emerging epidemics. Dorfman’s method, a classic group testing technique, helps reduce the number of tests required by pooling the samples of multiple individuals into a single sample for analysis. Dorfman’s method does not consider the time dynamics or limits on testing capacity involved in infection detection, and it assumes that individuals are infected independently, ignoring community correlations. To address these limitations, we present an adaptive group testing (AGT) strategy based on graph partitioning, which divides a physical contact network into subgraphs (groups of individuals) and assigns testing priorities based on the social contact characteristics of each subgraph. Our AGT aims to maximize the number of infected individuals detected and minimize the number of tests required. After each testing round (perhaps on a daily basis), the testing priority is increased for each neighboring group of known infected individuals. We also present an enhanced infectious disease transmission model that simulates the dynamic spread of a pathogen and evaluate our AGT strategy using the simulation results. When applied to 13 social contact networks, AGT demonstrates significant performance improvements compared to Dorfman’s method and its variations. Our AGT strategy requires fewer tests overall, reduces disease spread, and retains robustness under changes in group size, testing capacity, and other parameters. Testing plays a crucial role in containing and mitigating pandemics by identifying infected individuals and helping to prevent further transmission in families and communities. By identifying infected individuals and helping to prevent further transmission in families and communities, our AGT strategy can have significant implications for public health, providing guidance for policymakers trying to balance economic activity with the need to manage the spread of infection.

On the maximum number of edges in -critical graphs

Abstract A graph is called $k$ -critical if its chromatic number is $k$ but every proper subgraph has chromatic number less than $k$ . An old and important problem in graph theory asks to determine the maximum number of edges in an $n$ -vertex $k$ -critical graph. This is widely open for every integer $k\geq 4$ . Using a structural characterisation of Greenwell and Lovász and an extremal result of Simonovits, Stiebitz proved in 1987 that for $k\geq 4$ and sufficiently large $n$ , this maximum number is less than the number of edges in the $n$ -vertex balanced complete $(k-2)$ -partite graph. In this paper, we obtain the first improvement in the above result in the past 35 years. Our proofs combine arguments from extremal graph theory as well as some structural analysis. A key lemma we use indicates a partial structure in dense $k$ -critical graphs, which may be of independent interest.

Hypergraph partitioning using tensor eigenvalue decomposition.

Hypergraphs have gained increasing attention in the machine learning community lately due to their superiority over graphs in capturing super-dyadic interactions among entities. In this work, we propose a novel approach for the partitioning of k-uniform hypergraphs. Most of the existing methods work by reducing the hypergraph to a graph followed by applying standard graph partitioning algorithms. The reduction step restricts the algorithms to capturing only some weighted pairwise interactions and hence loses essential information about the original hypergraph. We overcome this issue by utilizing tensor-based representation of hypergraphs, which enables us to capture actual super-dyadic interactions. We extend the notion of minimum ratio-cut and normalized-cut from graphs to hypergraphs and show that the relaxed optimization problem can be solved using eigenvalue decomposition of the Laplacian tensor. This novel formulation also enables us to remove a hyperedge completely by using the "hyperedge score" metric proposed by us, unlike the existing reduction approaches. We propose a hypergraph partitioning algorithm inspired from spectral graph theory and also derive a tighter upper bound on the minimum positive eigenvalue of even-order hypergraph Laplacian tensor in terms of its conductance, which is utilized in the partitioning algorithm to approximate the normalized cut. The efficacy of the proposed method is demonstrated numerically on synthetic hypergraphs generated by stochastic block model. We also show improvement for the min-cut solution on 2-uniform hypergraphs (graphs) over the standard spectral partitioning algorithm.

EdgeMap: An Optimized Mapping Toolchain for Spiking Neural Network in Edge Computing.

Spiking neural networks (SNNs) have attracted considerable attention as third-generation artificial neural networks, known for their powerful, intelligent features and energy-efficiency advantages. These characteristics render them ideally suited for edge computing scenarios. Nevertheless, the current mapping schemes for deploying SNNs onto neuromorphic hardware face limitations such as extended execution times, low throughput, and insufficient consideration of energy consumption and connectivity, which undermine their suitability for edge computing applications. To address these challenges, we introduce EdgeMap, an optimized mapping toolchain specifically designed for deploying SNNs onto edge devices without compromising performance. EdgeMap consists of two main stages. The first stage involves partitioning the SNN graph into small neuron clusters based on the streaming graph partition algorithm, with the sizes of neuron clusters limited by the physical neuron cores. In the subsequent mapping stage, we adopt a multi-objective optimization algorithm specifically geared towards mitigating energy costs and communication costs for efficient deployment. EdgeMap-evaluated across four typical SNN applications-substantially outperforms other state-of-the-art mapping schemes. The performance improvements include a reduction in average latency by up to 19.8%, energy consumption by 57%, and communication cost by 58%. Moreover, EdgeMap exhibits an impressive enhancement in execution time by a factor of 1225.44×, alongside a throughput increase of up to 4.02×. These results highlight EdgeMap's efficiency and effectiveness, emphasizing its utility for deploying SNN applications in edge computing scenarios.

GraPH: Graph Partitioning Based on Hotspots

GraPH: Graph Partitioning Based on Hotspots

A novel approach of privacy protection of mobile users while using location-based services applications

The necessity of protecting mobile users' privacy when using location-based services (LBS) is highlighted in recent study, and location cloaking is one of the solutions offered. Location cloaking makes an effort to safeguard users' location privacy by introducing uncertainty while finding out the users’ location-context. The existing location cloaking algorithms are cautious about quality of the query service metrics (QoS), which mainly include success rate, anonymity level and spatial resolution. However, performance metric that is time to generate the cloaked regions should be improved. This was particularly important when user density is high. This work provides an efficient location cloaking scheme where a clique based cloaked region is constructed via partitioning of user-graph using graph partitioning algorithm, thereby generates reasonably-sized cloaked regions in less time. Further, to optimize cloaked region generation, multi-depth partitioning algorithm called BLH (Balanced Location Hiding) is proposed. Through concurrent and optimal generations of cloaked regions the proposed location cloaking preserves users' location privacy to a higher extent while retaining both the QoS and performance criteria. Experimental findings demonstrate that the success rate of the proposed algorithms is significantly increased. Even with a higher user density, it is approximately 98% with quicker response time.

Quantum annealing solution for the unrelated parallel machine scheduling with priorities and delay of task switching on machines

Quantum computing has emerged in recent years as an alternative to classical computing, which could improve the latter in solving some types of problems. One of the quantum programming models, Adiabatic Quantum Computing, has been successfully used to solve problems such as graph partitioning, traffic routing, and task scheduling. In this paper, the focus is on the scheduling of the problem of unrelated parallel machines, where the processing time of tasks on any of the available processing elements is known. Moreover, the proposed model is extended in two relevant aspects for this kind of problem: the existence of some degree of priority of tasks, and the introduction of a delay or penalty every time a processing unit or machine changes the type of task that executes.In all cases, the problem is expressed as Quadratic Unconstrained Binary Optimization, which can be subsequently solved using quantum annealers. The quantum nonlinear programming framework discussed in this work consists of three steps: quadratic approximation of cost function, a binary representation of parameter space, and solving the resulting Quadratic Unconstrained Binary Optimization on the quantum annealer platform D-Wave. One of the novelties in tackling this problem is the compaction of the model bearing in mind the repetitions of each task, to allow solving larger scheduling problems with the quantum resources available in the experimentation platform. An estimation of the number of qubits required in relation to the scheduling parameters is analyzed. The models have been implemented on the D-Wave platform and validated with respect to other traditional methods. Furthermore, the proposed extensions to consider priorities and to switch the delay of tasks have been analyzed using a case study.

Critical Path Awareness Techniques for Large-Scale Graph Partitioning

Graph partitioning is one of the fundamental problems in many graph-based applications and systems. It enables the division of a graph into smaller sub-graphs for subsequent parallel processing, reducing the processing latency of the graph. The critical path of a graph is the logical path with the longest delay from input to output. The processing time of the graph mainly depends on the delay incurred by the critical path, independent of other paths with small delays. Therefore, it can reduce the processing time of the graph by protecting the critical path of the graph from partition. However, existing approaches to graph partitioning only focus on metrics such as minimum cut and partition balance. As a result, the critical paths of graphs may be destroyed in the partitioning procedure. To address this problem, we present a critical path awareness approach, namely path-metis, to protect the critical paths and alleviate the processing latency after graph partitioning. In path-metis, two efficient strategies, including Slack and critical path fix strategies, are introduced. The Slack strategy, which incorporates critical path information into the weights of DAG, is used as pre-processing before traditional multi-level partitioning methods, like Metis. Then, for the generated partitioning scheme, the critical path fix strategy is proposed to further protect critical paths from being cut. We demonstrate the effectiveness of our approach on both real and synthetic datasets. From the experimental results, compared to Metis, our method improves critical path performance by 17.70%.

Lotan: Bridging the Gap between GNNs and Scalable Graph Analytics Engines

Recent advances in Graph Neural Networks (GNNs) have changed the landscape of modern graph analytics. The complexity of GNN training and the scalability challenges have also sparked interest from the systems community, with efforts to build systems that provide higher efficiency and schemes to reduce costs. However, we observe that many such systems basically "reinvent the wheel" of much work done in the database world on scalable graph analytics engines. Further, they often tightly couple the scalability treatments of graph data processing with that of GNN training, resulting in entangled complex problems and systems that often do not scale well on one of those axes. In this paper, we ask a fundamental question: How far can we push existing systems for scalable graph analytics and deep learning (DL) instead of building custom GNN systems? Are compromises inevitable on scalability and/or runtimes? We propose Lotan, the first scalable and optimized data system for full-batch GNN training with decoupled scaling that bridges the hitherto siloed worlds of graph analytics systems and DL systems. Lotan offers a series of technical innovations, including re-imagining GNN training as query plan-like dataflows, execution plan rewriting, optimized data movement between systems, a GNN-centric graph partitioning scheme, and the first known GNN model batching scheme. We prototyped Lotan on top of GraphX and PyTorch. An empirical evaluation using several real-world benchmark GNN workloads reveals a promising nuanced picture: Lotan significantly surpasses the scalability of state-of-the-art custom GNN systems, while often matching or being only slightly behind on time-to-accuracy metrics in some cases. We also show the impact of our system optimizations. Overall, our work shows that the GNN world can indeed benefit from building on top of scalable graph analytics engines. Lotan's new level of scalability can also empower new ML-oriented research on ever-larger graphs and GNNs.

Improving primary-vertex reconstruction with a minimum-cost lifted multicut graph partitioning algorithm

Particle physics experiments often require the simultaneous reconstruction of many interaction vertices. Usually, this problem is solved by ad hoc heuristic algorithms. We propose a universal approach to address the multiple vertex finding through a principled formulation as a minimum-cost lifted multicut problem. The suggested algorithm is tested in a typical LHC environment with multiple proton-proton interaction vertices. Reconstruction errors caused by the particle detectors complicate the solution and require the introduction of special metrics to assess the vertex-finding performance. We demonstrate that the minimum-cost lifted multicut approach outperforms heuristic algorithms and works well up to the highest vertex multiplicity expected at the LHC.

Exploring Tradeoffs in Automated School Redistricting: Computational and Ethical Perspectives

The US public school system is administered by local school districts. Each district comprises a set of schools mapped to attendance zones which are annually assessed to meet enrollment objectives. To support school officials in redrawing attendance boundaries, existing approaches have proven promising but still suffer from several challenges, including: 1) inability to scale to large school districts, 2) high computational cost of obtaining compact school attendance zones, and 3) lack of discussion on quantifying ethical considerations underlying the redrawing of school boundaries. Motivated by these challenges, this paper approaches the school redistricting problem from both computational and ethical standpoints. First, we introduce a practical framework based on sampling methods to solve school redistricting as a graph partitioning problem. Next, the advantages of adopting a modified objective function for optimizing discrete geometry to obtain compact boundaries are examined. Lastly, alternative metrics to address ethical considerations in real-world scenarios are formally defined and thoroughly discussed. Our findings highlight the inclusiveness and efficiency advantages of the designed framework and depict how tradeoffs need to be made to obtain qualitatively different school redistricting plans.

Task-Specific Scene Structure Representations

Understanding the informative structures of scenes is essential for low-level vision tasks. Unfortunately, it is difficult to obtain a concrete visual definition of the informative structures because influences of visual features are task-specific. In this paper, we propose a single general neural network architecture for extracting task-specific structure guidance for scenes. To do this, we first analyze traditional spectral clustering methods, which computes a set of eigenvectors to model a segmented graph forming small compact structures on image domains. We then unfold the traditional graph-partitioning problem into a learnable network, named Scene Structure Guidance Network (SSGNet), to represent the task-specific informative structures. The SSGNet yields a set of coefficients of eigenvectors that produces explicit feature representations of image structures. In addition, our SSGNet is light-weight (56K parameters), and can be used as a plug-and-play module for off-the-shelf architectures. We optimize the SSGNet without any supervision by proposing two novel training losses that enforce task-specific scene structure generation during training. Our main contribution is to show that such a simple network can achieve state-of-the-art results for several low-level vision applications including joint upsampling and image denoising. We also demonstrate that our SSGNet generalizes well on unseen datasets, compared to existing methods which use structural embedding frameworks. Our source codes are available at https://github.com/jsshin98/SSGNet.

A graphic partition method based on nodes learning for energy pipelines network simulation

The network of energy pipelines is extremely important for the proper functioning of a modern society. Even though measurement instruments and sensors are currently widely used to monitor the network’s operational status in real time, it is expensive to deploy these extra devices for a large city-scale network. Numerical simulation of fluid networks offers a rather easy and effective technique to track the work performance of pipelines. Because of the computational intensiveness of the pipeline network, it is most suitable to deploy such a massive simulation program on a supercomputer with a parallel technique. The performance of this kind of parallel simulation may undoubtedly be enhanced if the pipeline network is reasonably divided into several subnetworks in which the fluid flows are resolved independently. In this paper, we concentrate on how to achieve a computationally balanced subnetwork partition scheme and propose an acceleration method for energy pipeline network simulations based on a graph partition algorithm. In our approach, each node of the network takes part in the decision of dividing the entire network by learning the weight of its neighbors. Moreover, we utilize graph-tree transformations to merge locally related components as much as possible, and together with subgraph rebalancing, the pipeline network division quality is improved considerably. Numerical simulations show that our approach is much better than others for the degree of balance, and the cut edge ratio is also lower than others when it is compared with the random, k-means, and METIS methods.

Balanced connected partitions of graphs: approximation, parameterization and lower bounds

Balanced connected partitions of graphs: approximation, parameterization and lower bounds

SpecMap: An efficient spectral partitioning based static application mapping algorithm for 2D mesh NoCs

SummaryNetwork‐on‐chip (NoC) is adopted as a flexible and effective communication backbone by multiprocessor systems with core counts ranging from a few to hundreds. The performance of NoC based systems greatly rely on the communication between cores when two dependent tasks of an application are mapped to these cores. Hence, application mapping becomes a critical issue in NoC‐based systems as it affects the overall performance of the system. Application mapping aims to reduce the communication among the cores by carefully mapping the highly communicating tasks. Most of the existing static mapping approaches consider only single‐task platforms. In this article, we propose an efficient multi‐task static mapping algorithm called , for regular 2D NoC systems. consists of three stages namely, (i) graph partitioning, (ii) topology clustering, and (iii) mapping. In the graph partitioning stage, accurately identifies and groups the highly communicating tasks based on spectral graph partitioning. Then, the clustering and mapping stages allocate the most communicating tasks in the same or adjacent multi‐tasking cores. It also eliminates the interference between multiple applications by mapping them into different clusters of the NoC network. Simulations of have been conducted on different synthetic and real‐life applications using Gem5 simulator. The simulation results show considerable improvements for SpecMap in terms of communication cost, average network latency, energy consumption, throughput, and execution time as compared to the state‐of‐the‐art algorithms.

On the Minimum Bisection of Random 3-Regular Graphs

In this paper we give new bounds on the bisection width of random 3-regular graphs on $n$ vertices. The main contribution is a new lower bound of $0.103295n$ based on a first moment method together with a structural analysis of the graph, thereby improving a 27-year-old result of Kostochka and Melnikov. We also give a complementary upper bound of $0.139822n$ by combining a result of Lyons with original combinatorial insights. Developping this approach further, we obtain a non-rigorous improved upper bound with the help of Monte Carlo simulations.